If the sum of these integers is 67, how many integers are in the set?
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But not all integers plugged into this expression generate a prime number, and as integers get bigger, prime numbers become rarer.
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The rational numbers include the integers and any number that can be expressed as a ratio of two integers, such as 1, –4 and 99/100.
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The answer to the question from last year: 67 integers.
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Seeing as though you're interested, it asks:Is it possible to color all the integers either red or blue so that no pythagorean triple of integers a, b, c, satisfying a2+b2=c2 are all the same color?
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"There is an ancient analogy between integers and polynomials, which allows you to transform problems about integers, which are potentially very difficult, into problems about polynomials, which are also potentially difficult, but possibly more tractable," Shusterman said.
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But not all integers plugged into this expression generate a prime number.
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It's an EXPONENT because they sit high up next to the integers.
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The translation from integers to elliptic curves is a common one in mathematics.
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The other thing that's important about distance running is that it's such an integers-based sport.
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The conjecture predicts that there are infinitely many such pairs among the counting numbers, or integers.
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How old is David now?" and "How many three-digit integers can be written using only odd digits?
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As integers get bigger, prime numbers become rarer, but there is always a bigger prime number to be found.
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Mathematicians try to answer arithmetic questions over finite fields, and then hope to translate the results to the integers.
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For example, define a set of numbers such as the integers, all the whole numbers from minus infinity to positive infinity.
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Number theory is, in large part, the study of finding solutions to equations that cast insight into the fundamental properties of integers.
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For example, one problem I saw a lot was: Keep track of the k-th largest element in a stream of integers.
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According to Mr. Guzzetta, INTEGERS are "always" NATURAL or WHOLE numbers, "sometimes" RATIONAL (I know how they feel) numbers and "never" IMAGINARY numbers.
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There is no such thing as half a hole, the topologist would note, and the number of holes only changes stepwise in integers.
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Let's try — Dodging 9s The goal is to find the longest arithmetic progression of positive integers, none of whose terms contain the digit 20153.
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But you're trying to weigh yourself with these integers from year to year, based on how you feel, and the group you're running with.
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As a professor at Princeton, Dr. Langlands started investigating ideas that connected the mathematics of integers with a generalization of the theory of periodic functions.
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The resulting paper used the superposition of waves to settle a question about integers — a problem that had been intractable for more than 80 years.
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The abc conjecture describes the relationship between the three numbers in perhaps the simplest possible equation: a + b = c, for positive integers a, b and c.
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As opposed to integers or whole numbers, floating point numbers—with decimal points—are crucial to the calculations running through the neural networks involved with deep learning.
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With 32 bits, or 212 digits, we can represent a huge range of whole numbers (integers or ints, in computer science), all the way up to 21000.
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But types (integers, character strings, etc.), classes (modular units of related code), and methods (modular units that perform a specific function) are all referred to by emoji.
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Finite fields burst into prominence in the 23s, when André Weil devised a precise way of translating arithmetic in small number systems to arithmetic in the integers.
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It just means that I do not remember enough of what I might or might not have learned about integers in school to make much sense of the theme.
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Let S be the area of P. An odd positive integer n is given such that the squares of the side lengths of P are integers divisible by n.
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Prime numbers are integers (whole numbers) that can only be divided by themselves or the number 1, and they appear along the number line in a highly erratic way.
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Mathematicians endeavored for centuries to prove Fermat's assertion that an + bn = cn is impossible where a, b, c and n are positive integers and n is greater than 2.
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Fermat asserted that equations of the form aⁿ + bⁿ = cⁿ do not have solutions when n is an integer greater than 2 and a, b and c are positive integers.
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Their appearance in the roll call of all integers cannot be predicted, and no magical formula exists to know when a prime number will choose to suddenly make an appearance.
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Any of the Steinitz-type proofs will tell you not only that there is a polyhedron but also that there's a polyhedron with integers for the coordinates of the vertices.
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Running Shor's algorithm for integers large enough to unlock a standard 22016-bit encryption key, for instance, would require thousands of qubits—plus probably many thousands more to correct for errors.
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Leonardo Fibonacci, meanwhile, was an early modern Italian mathematician who helped popularize the Hindu–Arabic numeral system as well as a series of integers that became known as the Fibonacci sequence.
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Nonetheless, the sprawling cast of characters in this populous drama, blandly directed by Doug Hughes, registers as so many integers, waiting to be clicked into prearranged places in a socio-economic equation.
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It also stands in contrast to what central banks around the globe have been doing: cutting rates into negative integers and increasing quantitative easing operations to fend off a global economic downturn.
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Piper introduces the dissertation, "The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist's Rendering," with the following: A fascinating tale of mayhem, mystery and mathematics.
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"This is the first work that gives a quantitative analogue of what is expected to be true over the integers, and that is something that really stands out," said Zeev Rudnick of Tel Aviv University.
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It's a freaky quirk of the Universe that has captivated mathematicians for centuries, as their appearance in the roll call of all integers defies prediction (though biases have been detected in the distribution of consecutive primes).
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So whether your video is 24hz, 25hz, 30hz, 48hz, 50hz, 60hz, 90hz, 100hz, or 120hz, you'll be able to use Flicks to ensure that everything is in sync while still using whole integers (instead of decimals).
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Pierre de Fermat, a 17th century French mathematician, had asserted that equations of the form an + bn = cn, where a, b, c and n are positive integers, have no solutions when n is greater than two.
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In the 1940s, Otto E. Neugebauer and Abraham J. Sachs, mathematics historians, pointed out that the other three columns were essentially Pythagorean triples — sets of integers, or whole numbers, that satisfy the equation a2 + b0003 = c2.
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Contestants can use only a pencil and paper and have 45 seconds to solve word problems such as this one answered by the winner in 2014: The smallest integer of a set of consecutive integers is -32.
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Most famously, in 1994, a young staffer at Bell Laboratories named Peter Shor proposed a quantum algorithm that factors integers exponentially faster than any known classical algorithm—an efficiency that could allow it to crack many popular encryption schemes.
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The trio of researchers have already claimed the reward..Turns out that the answer to the puzzle is: No. But to reach that simple conclusion, the team had to work through combinations of integers all the way up to 7,825.
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For a sufficiently large subset of integers — one that mathematicians describe as having a positive density — it is possible to find arbitrarily long arithmetic progressions, which are sequences like 3, 7, 19913, 15 where the numbers are equally spaced apart.
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Whereas the abc conjecture describes an underlying mathematical phenomenon in terms of relationships between integers, Szpiro's conjecture casts that same underlying relationship in terms of elliptic curves, which give a geometric form to the set of all solutions to a type of algebraic equation.
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In 1986, Kenneth Ribet of the University of California, Berkeley, proved an intriguing connection: If Fermat's Last Theorem were wrong, and there indeed existed a set of integers that fit the equation, that would generate an elliptic curve that violated the Taniyama-Shimura conjecture.
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In 2004, Dr. Tao and Ben Green, a mathematician at the University of Oxford, cited Dr. Furstenberg and used ergodic theory arguments to prove a major result — that arbitrarily long progressions also exist among the prime numbers, the integers that have exactly two divisors: 1 and themselves.
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The conjecture appeared unconnected to Fermat's Last Theorem, a seemingly simple statement made by the French mathematician Pierre de Fermat in 21960: Equations of the form an + bn = cn do not have solutions when n is an integer greater than 270 and a, b and c are positive integers.
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The three numbers a, b and c are supposed to be positive integers, and they are not allowed to share any common prime factors — so, for example, we could consider the equation 8 + 93 = 17, or 5 + 16 = 21, but not 6 + 9 = 15, since 6, 9 and 83 are all divisible by 3.
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In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of integers, such as , and the complex number , which generates the Gaussian integers.
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The symbol can be annotated to denote various sets, with varying usage amongst different authors: , or for the positive integers, or for non-negative integers, and for non-zero integers. Some authors use for non-zero integers, while others use it for non-negative integers, or for }. Additionally, is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.
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The densest lattice packing of unit spheres in four dimensions (called the D4 lattice) places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to the Hurwitz integers: quaternions whose real coefficients are either all integers or all half-integers..
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The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If is a number field, its ring of integers is the subring of algebraic integers in , and is frequently denoted as . These are the prototypical examples of Dedekind domains.
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For example, the -adic integers are the ring of integers of the -adic numbers .
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Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer) An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are , and . Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers. The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
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Storage of integers is usually done with dedicated 32 or 64 bits per integer. For small integers, packing multiple integers into the same space makes storage more efficient.
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Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.
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This scheme also uses special sequences of integers. Let be integers. We consider a sequence of pairwise coprime positive integers m_0 < ... < m_n such that m_0.m_{n-k+2}...m_n < m_1...m_k.
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Half-integers occur frequently enough in mathematics that a distinct term is convenient. Note that a halving an integer does not always produce a half- integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a special case of the dyadic rationals (numbers produced by dividing an integer by a power of two)..
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Elias omega coding does not code zero or negative integers. One way to code all non negative integers is to add 1 before coding and then subtract 1 after decoding. One way to code all integers is to set up a bijection, mapping all integers (0, 1, -1, 2, -2, 3, -3, ...) to strictly positive integers (1, 2, 3, 4, 5, 6, 7, ...) before coding.
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The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both rational. The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals. This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation :x^2 +cx+d=0, with and integers.
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The integers are a set of numbers commonly encountered in arithmetic and number theory. There are many subsets of the integers, including the natural numbers, prime numbers, perfect numbers, etc. Many integers are notable for their mathematical properties. Notable integers include −1, the additive inverse of unity, and 0, the additive identity.
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In particular, even integers are exactly those integers k where This formulation is useful for investigating integer zeroes of polynomials.
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Elias delta coding does not code zero or negative integers. One way to code all non negative integers is to add 1 before coding and then subtract 1 after decoding. One way to code all integers is to set up a bijection, mapping integers all integers (0, 1, −1, 2, −2, 3, −3, ...) to strictly positive integers (1, 2, 3, 4, 5, 6, 7, ...) before coding. This bijection can be performed using the "ZigZag" encoding from Protocol Buffers (not to be confused with Zigzag code, nor the JPEG Zig- zag entropy coding).
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Remember to use integers, otherwise much quantization error will become involved, as we previously quantized everything to integers in the encoder.
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The set of all integers, {..., -1, 0, 1, 2, ...} is a countably infinite set. The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers. The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers.
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We also assume that always denotes a finite sequence of non-negative integers and we will use the notation: : and . Observe that for any integers and , :. From this it follows that if consists of distinct positive integers then . We show by induction on that if consists of non-negative integers such that for some integer then .
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Spectrum: HZp (Eilenberg–Maclane spectrum of the integers mod p.) Coefficient ring: πn(HZp) = Zp (Integers mod p) if n = 0, 0 otherwise.
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Indeed, this is the usual construction to obtain the integers from the natural numbers. See "Construction" under Integers for a more detailed explanation.
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Bijective mapping from integer to even numbers To understand what this means, we first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that for every integer, there is a distinct even integer: ... −2→−4, −1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n (see picture).
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In the ring of integers (on real numbers), if is a unit, then is either 2 or 0. But are the usual Mersenne primes, and the formula does not lead to anything interesting (since it is always −1 for all ). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.
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One defines the ring of integers of a non-archimedean local field as the set of all elements of with absolute value ; this is a ring because of the strong triangle inequality.Cassels (1986) p. 41 If is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.
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Consider two even integers and . Since they are even, they can be written as : x =2a : y=2b respectively for integers and . Then the sum can be written as : x+y = 2a + 2b = 2(a+b)=2p where p=a+b, and are all integers. It follows that has 2 as a factor and therefore is even, so the sum of any two even integers is even.
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In mathematics, a lattice word (or lattice permutation) is a string composed of positive integers, in which every prefix contains at least as many positive integers i as integers i \+ 1. A reverse lattice word, or Yamanouchi word, is a string whose reversal is a lattice word.
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The classification of p-compact groups states that there is a 1-1 correspondence between connected p-compact groups, and root data over the p-adic integers. This is analogous to the classical classification of connected compact Lie groups, with the p-adic integers replacing the rational integers.
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It is likely that Inca mathematics at least allowed division of integers into integers or fractions and multiplication of integers and fractions. According to mid-17th-century Jesuit chronicler Bernabé Cobo,Cobo, B. (1983 [1653]). Obras del P. Bernabé Cobo. Vol. 1. Edited and preliminary study By Francisco Mateos.
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The proof uses Euclid's lemma (Elements VII, 30): If a prime divides the product of two integers and , then must divide at least one of those integers and .
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Version 1 NRG format stores values as 32-bit integers. Nero Burning ROM v5.5 introduced a new NRG file format, version 2, with support for 64-bit integers.
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Observe that for any integers and , :. From this it follows that if consists of distinct positive integers then . We show by induction on that if consists of non-negative integers such that for some integer then . This is clearly true for and so assume that , which implies that all are positive.
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For the integers and and × is said to be distributive over +. The integers are an example of a ring. The integers have additional properties which make it an integral domain. A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×.
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84 The numbers built up from a cube root of unity are now called the ring of Eisenstein integers. The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
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Distribution of Gaussian primes in the complex plane, with norms less than 500 The Gaussian integers are complex numbers of the form , where and are ordinary integersThe phrase "ordinary integer" is commonly used for distinguishing usual integers from Gaussian integers, and more generally from algebraic integers. and is the square root of negative one. By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above. Reprinted in and This unique factorization is helpful in many applications, such as deriving all Pythagorean triples or proving Fermat's theorem on sums of two squares.
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In analyzing the performance of binary search, another consideration is the time required to compare two elements. For integers and strings, the time required increases linearly as the encoding length (usually the number of bits) of the elements increase. For example, comparing a pair of 64-bit unsigned integers would require comparing up to double the bits as comparing a pair of 32-bit unsigned integers. The worst case is achieved when the integers are equal.
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Another common example is the non-real cubic root of unity , which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
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If two integers are equal, then all their residues are equal. Conversely, if all residues are equal, then the two integers are equal, or their differences is a multiple of . It follows that testing equality is easy. At the opposite, testing inequalities () is difficult and, usually, requires to convert integers to the standard representation.
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This representation is no longer bijective, as the entire set of left-infinite sequences of digits is used to represent the k-adic integers, of which the integers are only a subset.
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Plot of a linear Diophantine equation, 9x + 12y = 483\. The solutions are shown as blue circles. Diophantine equations are equations in which the solutions are restricted to integers; they are named after the 3rd- century Alexandrian mathematician Diophantus. A typical linear Diophantine equation seeks integers x and y such that : where a, b and c are given integers.
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The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.
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Elias γ code or Elias gamma code is a universal code encoding positive integers developed by Peter Elias. It is used most commonly when coding integers whose upper-bound cannot be determined beforehand.
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Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group Z/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set with addition modulo 2; informally, it is sometimes said that Z/2Z equals the set with addition modulo 2.
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The usual numbers systems used in numerical computation are floating point numbers and integers of a fixed bounded size. None of these is convenient for computer algebra, due to expression swell. Therefore, the basic numbers used in computer algebra are the integers of the mathematicians, commonly represented by an unbounded signed sequence of digits in some base of numeration, usually the largest base allowed by the machine word. These integers allow to define the rational numbers, which are irreducible fractions of two integers.
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However, applications that regularly handle integers wider than 32 bits, such as cryptographic algorithms, will need a rewrite of the code handling the huge integers in order to take advantage of the 64-bit registers.
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In contrast, a discrete variable over a particular range of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The number of permitted values is either finite or countably infinite. Common examples are variables that must be integers, non- negative integers, positive integers, or only the integers 0 and 1. Methods of calculus do not readily lend themselves to problems involving discrete variables.
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A set of integers can also be called coprime if its elements share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that and are coprime for every pair of different integers in the set. The set } is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as . This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.
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Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder, such that a = bq + r and 0 ≤ r < , where denotes the absolute value of b.
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Or, suppose we are given a finite set of nonzero integers, and are asked to mark as large a subset as possible of this set under the restriction that the sum of any two marked integers cannot be marked. It appears that (independent of what the given integers actually are!) we can always mark at least one-third of them.
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In 1896, Williamina Fleming observed mysterious spectral lines from Zeta Puppis, which fit the Rydberg formula if half-integers were used instead of whole integers. It was later found that these were due to ionized helium.
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Pentium II processor with MMX technology MMX defines eight registers, called MM0 through MM7, and operations that operate on them. Each register is 64 bits wide and can be used to hold either 64-bit integers, or multiple smaller integers in a "packed" format: a single instruction can then be applied to two 32-bit integers, four 16-bit integers, or eight 8-bit integers at once. MMX provides only integer operations. When originally developed, for the Intel i860, the use of integer math made sense (both 2D and 3D calculations required it), but as graphics cards that did much of this became common, integer SIMD in the CPU became somewhat redundant for graphical applications.
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In order to explain the verifier- based definition of NP, consider the subset sum problem: Assume that we are given some integers, {−7, −3, −2, 5, 8}, and we wish to know whether some of these integers sum up to zero. Here, the answer is "yes", since the integers {−3, −2, 5} corresponds to the sum The task of deciding whether such a subset with zero sum exists is called the subset sum problem. To answer if some of the integers add to zero we can create an algorithm which obtains all the possible subsets. As the number of integers that we feed into the algorithm becomes larger, both the number of subsets and the computation time grows exponentially.
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There are also quadratic reciprocity laws in rings other than the integers.
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Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. The final nonzero remainder is , the Gaussian integer of largest norm that divides both and ; it is unique up to multiplication by a unit, or . Many of the other applications of the Euclidean algorithm carry over to Gaussian integers. For example, it can be used to solve linear Diophantine equations and Chinese remainder problems for Gaussian integers; continued fractions of Gaussian integers can also be defined.
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In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers). The set of all algebraic integers, , is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring is the integral closure of regular integers in complex numbers. The ring of integers of a number field , denoted by , is the intersection of and : it can also be characterised as the maximal order of the field .
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In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.Cupillari, p. 20. For example, direct proof can be used to prove that the sum of two even integers is always even: :Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for integers a and b.
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One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below.
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Van der Waerden's theorem states that for any positive integers r and k there exists a positive integer N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The smallest such N is the van der Waerden number W(r, k).
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Since the even integers form a subgroup of the integers, they partition the integers into cosets. These cosets may be described as the equivalence classes of the following equivalence relation: if is even. Here, the evenness of zero is directly manifested as the reflexivity of the binary relation ~. There are only two cosets of this subgroup—the even and odd numbers—so it has index 2.
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This is analogous to the relationship between the rational numbers and the integers.
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Here n, k, and r are non-negative integers whose sum is even.
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In mathematics, a monothetic group is a topological group with a dense cyclic subgroup. They were introduced by . An example is the additive group of p-adic integers, in which the integers are dense. A monothetic group is necessarily abelian.
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In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.. The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero..
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Prove that is a perfect square. # Fix some value that is a non-square positive integer. Assume there exist positive integers for which . # Let be positive integers for which and such that is minimized, and without loss of generality assume .
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Bézout's identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b. In other words, it is always possible to find integers s and t such that g = sa + tb. The integers s and t can be calculated from the quotients q0, q1, etc. by reversing the order of equations in Euclid's algorithm.
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Data types include integers, fix point, floating point, Hollerith character strings, status or Booleans.
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Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
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Common examples of square-free elements include square-free integers and square-free polynomials.
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For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal (6) in Z[] factors into prime ideals as : (6) = (2, 1 + )(2,1 − )(3, 1 + )(3, 1 − ). However, unlike Z as the ring of integers of Q, the ring of integers of a proper extension of Q need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in , the uniqueness of the factorization fails: : 6 = 2 ⋅ 3 = (1 + ) ⋅ (1 − ).
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Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.
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More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain , "rational numbers" must be replaced by "field of fractions of ". This implies that, if is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials.
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A knot diagram with crossings labelled for a Dowker sequence The Dowker–Thistlethwaite notation, also called the Dowker notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10).
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Given two integers a and b, the algorithm performs O(\log\ a + \log\ b) arithmetic operations on numbers with at most O(\log\ a + \log\ b) bits. At the same time, the number of arithmetic operations cannot be bounded by the number of integers in the input (which is constant in this case, there are always only two integers in the input). Due to the latter observation, the algorithm does not run in strongly polynomial time. Its real running time depends on the magnitudes of a and b and not only on the number of integers in the input.
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Concretely, transform the integer as `(n << 1) ^ (n >> k - 1)` for fixed k-bit integers.
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344 = 23 × 43, octahedral number, noncototient, totient sum of the first 33 integers, refactorable number.
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It's not mandatory to use all the numbers. All numbers used must be positive integers.
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Green and Tao's proof has three main components: # Szemerédi's theorem, which asserts that subsets of the integers with positive upper density have arbitrarily long arithmetic progressions. It does not a priori apply to the primes because the primes have density zero in the integers. # A transference principle that extends Szemerédi's theorem to subsets of the integers which are pseudorandom in a suitable sense. Such a result is now called a relative Szemerédi theorem.
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The reciprocal of is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as goes to infinity, the probability that three positive integers less than chosen uniformly at random will be relatively prime approaches this value).
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Consider the multiplicative monoid of positive integers as a category with one object. In this category, the pullback of two positive integers and is just the pair , where the numerators are both the least common multiple of and . The same pair is also the pushout.
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Yao showed that every nonempty set of nonnegative integers is the score set for some tournament.
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The most general data type for a rational number stores the numerator and denominator as integers.
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Sound Soldiers is an album by Gruvis Malt. It was released by Integers Only in 1999.
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An example of an infinite set is the set of positive integers {1, 2, 3, 4, ...}.
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The sum of the integers from 1 to 36 is 666 (see number of the beast).
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The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers.
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Indexed arrays are simply hashes using integers as keys. Objects can syntactically be used as Arrays.
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Consider the expression: :"The smallest positive integer not definable in under sixty letters." Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under sixty letters".
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Any computer programming language or software package that is used to compute D mod 97 directly must have the ability to handle integers of more than 30 digits. In practice, this can only be done by software that either supports arbitrary-precision arithmetic or that can handle 220 bit (unsigned) integers,The maximum length of D in (decimal) digits for the fully generic IBAN with 34 alphanumeric digits (two of which, the check digits, can, however, only be numeric) is . 2220 is equal to 1.7 × 1066, from which it can be inferred that 220 bit unsigned integers can accommodate all unsigned integers of 66 digits. features that are often not standard.
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The usual proof involves another lemma called Bézout's identity. This states that if and are relatively prime integers (i.e. they share no common divisors other than 1 and -1) there exist integers and such that : rx+sy = 1. Let and be relatively prime, and assume that .
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For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then :IO_L\ is a principal ideal αOL, for OL the ring of integers of L and some element α in it.
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The map P \to Q is slightly problematic, in the sense that the indices of the P-vertices are naturally odd integers whereas the indices of Q-vertices are naturally even integers. A more conventional approach to the labeling would be to label the vertices of P and Q by integers of the same parity. One can arrange this either by adding or subtracting 1 from each of the indices of the Q-vertices. Either choice is equally canonical.
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These concepts can even assist with in number- theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers., Section I.7, p.
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In this section, a ring can be viewed as merely an abstract set in which one can perform the operations of addition and multiplication; analogous to the integers. The ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the fundamental theorem of arithmetic. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds.
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This algorithm, as mentioned above, is very efficient for numbers of the form re±s, for r and s relatively small. It is also efficient for any integers which can be represented as a polynomial with small coefficients. This includes integers of the more general form are±bsf, and also for many integers whose binary representation has low Hamming weight. The reason for this is as follows: The Number Field Sieve performs sieving in two different fields.
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For example, the problem FIND-SUBSET-SUM is in NP-equivalent. Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero (or returning the empty set if there is no such subset). This optimization problem is similar to the decision problem SUBSET-SUM. Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.
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It is known, that the harmonic numbers are never integers except the case n=1. The same question can be posed with respect to the hyperharmonic numbers: are there integer hyperharmonic numbers? István Mező proved that if r=2 or r=3, these numbers are never integers except the trivial case when n=1. He conjectured that this is always the case, namely, the hyperharmonic numbers of order r are never integers except when n=1.
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These 7 maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders. These maximal orders were constructed by , Dickson and Bruck as follows. Label the 8 basis vectors by the points of the projective line over the field with 7 elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets :∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245) of the extended quadratic residue code of length 8 over the field of 2 elements, given by ∅, (∞124) and its images under adding a constant mod 7, and the complements of these 8 sets.
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Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de Méziriac, and it was solved by Joseph-Louis Lagrange in his four-square theorem in 1770, the same year Waring made his conjecture. Waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the fourth power, and so forth, to show that any positive integer may be represented as the sum of other integers raised to a specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way.
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A Young diagram representing visually a polite expansion 15 = 4 + 5 + 6 In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. Other positive integers are impolite... Polite numbers have also been called staircase numbers because the Young diagrams representing graphically the partitions of a polite number into consecutive integers (in the French style of drawing these diagrams) resemble staircases.... If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid (trapezium outside North America)........ The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester,. In The collected mathematical papers of James Joseph Sylvester (December 1904), H. F. Baker, ed. Sylvester defines the class of a partition into distinct integers as the number of blocks of consecutive integers in the partition, so in his notation a polite partition is of first class.
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It follows that the algebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ring contained in) F. A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of F is an integral domain. The field F is the field of fractions of the integral domain OF. This way one can get back and forth between the algebraic number field F and its ring of integers OF. Rings of algebraic integers have three distinctive properties: firstly, OF is an integral domain that is integrally closed in its field of fractions F. Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
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In particular, if a theory preserves CP symmetry then the electric charges of all dyons are integers.
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For instance, a generic `numeric` type might be supplied instead of integers of some specific bit- width.
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There are algorithms to solve all the problems addressed in this article over the integers. In other words, linear algebra is effective over the integers. See Linear Diophantine system for details. The same solution applies to the same problems over a principal ideal domain, with the following modifications.
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The equal sums problem is the following problem. Given n positive integers that sum to less than 2^n - 1, find two distinct subsets of the integers that have the same total. This problem is contained in PPP, but it is not known if it is PPP-complete.
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Every Robbins pentagon may be scaled so that its sides and area are integers. More strongly, Buchholz and MacDougall showed that if the side lengths are all integers and the area is rational, then the area is necessarily also an integer, and the perimeter is necessarily an even number.
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In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
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A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.
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Integers can be divided into even and odd, those that are evenly divisible by two and those that are not. The even integers are ...−4, −2, 0, 2, 4, whereas the odd integers are −3, −1, 1, 3,... The property of whether an integer is even (or not) is known as its parity. If two numbers are both even or both odd, they have the same parity. By contrast, if one is even and the other odd, they have different parity.
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If is a field, the polynomial ring has many properties that are similar to those of the ring of integers \Z. Most of these similarities result from the similarity between the long division of integers and the long division of polynomials. Most of the properties of that are listed in this section do not remain true if is not a field, or if one consider polynomials in several indeterminates. Like for integers, the Euclidean division of polynomials has a property of uniqueness.
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Euclidean division is based on the following result, which is sometimes called Euclid's division lemma. Given two integers and , with , there exist unique integers and such that : and :, where denotes the absolute value of . In the above theorem, each of the four integers has a name of its own: is called the , is called the , is called the and is called the . The computation of the quotient and the remainder from the dividend and the divisor is called or — in case of ambiguity — .
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The Lambek–Moser theorem is universal, in the sense that it can explain any partition of the integers into two infinite parts. If and are any two infinite subsets forming a partition of the integers, one may construct a pair of functions and from which this partition may be derived using the Lambek–Moser theorem: define and . For instance, consider the partition of integers into even and odd numbers: let be the even numbers and be the odd numbers. Then , so and similarly .
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Two integers whose difference is a multiple of have the same representation in the residue numeral system defined by the s. More precisely, the Chinese remainder theorem asserts that each of the different sets of possible residues represents exactly one residue class modulo . That is, each set of residues represents exactly one integer in the interval . In applications where one is also interested with negative integers, it is often more convenient to represent integers belonging to an interval centered at 0.
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The Furstenberg integers are separable and metrizable, but incomplete. By Urysohn's metrization theorem, they are regular and Hausdorff.
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Levenstein coding, or Levenshtein coding, is a universal code encoding the non-negative integers developed by Vladimir Levenshtein.
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Given a group G and a subgroup H, and an element a ∈ G, one can consider the corresponding left coset: aH := { ah : h ∈ H }. Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup H of even integers. Then there are exactly two cosets: 0 + H, which are the even integers, and 1 + H, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation). For a general subgroup H, it is desirable to define a compatible group operation on the set of all possible cosets, { aH : a ∈ G }.
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In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.
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Fibonacci, Elias Gamma, and Elias Delta vs binary coding Rice with k = 2, 3, 4, 5, 8, 16 versus binary In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic (i.e., p(i) ≥ p(i + 1) for all positive i), the expected lengths of the codewords are within a constant factor of the expected lengths that the optimal code for that probability distribution would have assigned. A universal code is asymptotically optimal if the ratio between actual and optimal expected lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity. In general, most prefix codes for integers assign longer codewords to larger integers.
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Using the corresponding strict relation "<", the open interval is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b. For example, the open interval on the integers is empty since there are no integers such that .
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The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
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A strengthening in a different direction, Szemerédi's theorem, shows that dense sets of integers contain arbitrarily long arithmetic progressions.
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Even–Rodeh code is a universal code encoding the non-negative integers developed by Shimon Even and Michael Rodeh.
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Elias δ code or Elias delta code is a universal code encoding the positive integers developed by Peter Elias.
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Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that x−1 = , meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b. Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.
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The current domain of a variable can be inspected using specific literals; for example, `dom(X,D)` finds out the current domain `D` of a variable `X`. As for domains of reals, functors can be used with domains of integers. In this case, a term can be an expression over integers, a constant, or the application of a functor over other terms. A variable can take an arbitrary term as a value, if its domain has not been specified to be a set of integers or constants.
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The termial was coined by Donald E. Knuth in his The Art of Computer Programming. It is the additive analog of the factorial function, which is the product of integers from to . He used it to illustrate the extension of the domain from positive integers to the real numbers.Donald E. Knuth (1997).
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However, only the first pipeline has a barrel shifter and hardware for confirming the prediction of conditional branches. The second pipeline is used to access the multiplier and divider. Multiplies are pipelined, and have a six-cycle latency for 32-bit integers and ten for 64-bit integers. Division is not pipelined.
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Then the sum x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence, the sum of any two even integers is even. This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and distributivity.
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The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer.
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W. Galway wrote a computer programme to determine odd integers not expressible as . Galway verified that there are only eighteen numbers less than not representable in the form . Based on Galway's computations, Ken Ono and K. Soundararajan formulated the following conjecture: :The odd positive integers which are not of the form x2 \+ are: .
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Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup.Knapp (2006), [ p. 170] One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order.
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The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer.
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A Proth number takes the form N=k 2^n +1 where k and n are positive integers, k is odd and 2^n>k. A Proth prime is a Proth number that is prime. Without the condition that 2^n > k, all odd integers larger than 1 would be Proth numbers.
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Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Dudeney numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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The equation has no non-trivial (i.e. ) solutions in integers. In fact, it has none in Eisenstein integers.Hardy & Wright, Thm.
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Kaprekar numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Suppose and are integers. Let be a ratio given in its lowest terms. Draw the arcs and with centre . Join .
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Self numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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The game is intended for Grades 2-8 and teaches operations involving whole numbers, integers, fractions, decimals, and rational numbers.
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2Z (blue) as subgroup of Z In abstract algebra, the even integers form various algebraic structures that require the inclusion of zero. The fact that the additive identity (zero) is even, together with the evenness of sums and additive inverses of even numbers and the associativity of addition, means that the even integers form a group. Moreover, the group of even integers under addition is a subgroup of the group of all integers; this is an elementary example of the subgroup concept. The earlier observation that the rule "even − even = even" forces 0 to be even is part of a general pattern: any nonempty subset of an additive group that is closed under subtraction must be a subgroup, and in particular, must contain the identity.
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Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.
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Before modern factoring algorithms, such as MPQS and NFS, were developed, it was thought to be useful to select Blum integers as RSA moduli. This is no longer regarded as a useful precaution, since MPQS and NFS are able to factor Blum integers with the same ease as RSA moduli constructed from randomly selected primes.
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Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.
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Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of Winning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers.
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The following image shows a simple APL \ 1130 session. This session was performed via the 1130 simulator available from IBM 1130.org apl \ 1130 apl \ 1130 sample session The above session shows a signon, addition of the integers 1 to 100, generation of an addition table for the integers 1..5 and a sign off.
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The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales.
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The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers, see construction of the real numbers.
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The theorem applies to any non-decreasing and unbounded function that maps positive integers to non-negative integers. From any such function , define to be the integer-valued function that is as close as possible to the inverse function of , in the sense that, for all , :. It follows from this definition that . Further, let : and .
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First, s can be divided into a vector of M 32-bit integers called vsecret. Then players are each given a vector of M random integers, player i receiving vi. The remaining player is given vn = (vsecret − v1 − v2 − ... − vn−1). The secret vector can then be recovered by summing across all the player's vectors.
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The norm of a Gaussian integer x + yi is the number x2 + y2. Thus, the Pythagorean primes (and 2) occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, because they can be factored as :p = (x + yi)(x − yi).
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Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers. This implies also that the factorization over the rationals of a polynomial with rational coefficients is the same as the factorization over the integers of its primitive part. Similarly, the factorization over the integers of a polynomial with integer coefficients is the product of the factorization of its primitive part by the factorization of its content.
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A diagram showing a representation of the equivalent classes of pairs of integers The rational numbers may be built as equivalence classes of ordered pairs of integers. More precisely, let be the set of the pairs of integers such . An equivalence relation is defined on this set by : if and only if . Addition and multiplication can be defined by the following rules: :\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right), :\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right).
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A good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as , will give another function with that property.
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The simplest linear Diophantine equation takes the form , where , and are given integers. The solutions are described by the following theorem: :This Diophantine equation has a solution (where and are integers) if and only if is a multiple of the greatest common divisor of and . Moreover, if is a solution, then the other solutions have the form , where is an arbitrary integer, and and are the quotients of and (respectively) by the greatest common divisor of and . Proof: If is this greatest common divisor, Bézout's identity asserts the existence of integers and such that .
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As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers). 1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.
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A rational number can be defined as the quotient of two integers (as long as the denominator is non-zero). A more detailed definition goes as follows: : A real number r is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational. Or more formally: : Given a real number r, r is rational if and only if there exists integers a and b such that r = \tfrac a b and b eq 0.
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Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color. The least such N is the Van der Waerden number W(r, k), named after the Dutch mathematician B. L. van der Waerden.
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First four pronic numbers as sums of the first n even numbers. The th pronic number is the sum of the first even integers. All pronic numbers are even, and 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number... The number of off-diagonal entries in a square matrix is always a pronic number.. The fact that consecutive integers are coprime and that a pronic number is the product of two consecutive integers leads to a number of properties.
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The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.
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In the case of the integers and Abelian groups a pure projective module amounts to a direct sum of cyclic groups.
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The digit sum can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences.
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Perfect digital invariants can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Refinements that use rational numbers instead of integers can avoid renumbering, and so are faster to update, although much more complicated.
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Sum-product numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.
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Certificate grades are either rounded to integers or half-integers. After having rounded the individual grades, a weighted mean is used to calculate the overall result. The weight of a grade is normally proportional to the number of hours the according subject was taught per week. To pass a year, this overall result needs to be sufficient.
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Integer return values (similar to x86) are returned in RAX if 64 bits or less. Floating point return values are returned in XMM0. Parameters less than 64 bits long are not zero extended; the high bits are not zeroed. Structs and unions with sizes that match integers are passed and returned as if they were integers.
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For this reason, Gauss's result is sometimes known as the Eureka theorem.. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813. The proof of is based on the following lemma due to Cauchy: For odd positive integers and such that and we can find nonnegative integers , , , and such that and .
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There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly 7 maximal orders containing it.
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A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group). Every finite subgroup of a cyclically ordered group is cyclic..
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However, the very possibility of implementing such an operator highly constrains the Number type (for example, one can't compare an integer with a complex number), and actually only comparing integers with integers and reals with reals makes sense. Rewriting this function so that it would only accept 'x' and 'y' of the same type requires bounded polymorphism.
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INTERCAL-72 (the original version of INTERCAL) had only four data types: the 16-bit integer (represented with a `.`, called a "spot"), the 32-bit integer (`:`, a "twospot"), the array of 16-bit integers (`,`, a "tail"), and the array of 32-bit integers (`;`, a "hybrid"). There are 65535 available variables of each type, numbered from `.1` to `.
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Notice that mn, dn, and an are always integers. The algorithm terminates when this triplet is the same as one encountered before.
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This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.
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Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Collection of articles in memory of Juriĭ Vladimirovič Linnik.
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The class C∞ of infinitely differentiable functions, is the intersection of the classes Ck as k varies over the non-negative integers.
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Eleanor Mollie Horadam (29 June 1921 – 5 May 2002) was an English-Australian mathematician specialising in the number theory of generalised integers.
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Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". See Andrew Odlyzko's website for their tables and bibliographies.
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Spectrum: H (Eilenberg–MacLane spectrum of the integers.) Coefficient ring: πn(H) = Z if n = 0, 0 otherwise. The original homology theory.
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It counts the number of positive integers less than or equal to that have at least one prime factor in common with .
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Then the result states that and are strictly increasing and that the ranges of and form a partition of the positive integers.
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The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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The frequency and placement of all integers and problems were counterbalanced, and no addend, augend, or sum was presented on consecutive trials.
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Given an unramified finite extension of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields. Suppose is an unramified extension of local fields, with ring of integers OK of such that the residue field, the integers of modulo their unique maximal ideal , is a finite field of order , where is a power of a prime. If is a prime of lying over , that is unramified means by definition that the integers of modulo , the residue field of , will be a finite field of order extending the residue field of where is the degree of . We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that :s_\Phi(x) \equiv x^q \mod \Phi.
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In number theory the n conjecture is a conjecture stated by as a generalization of the abc conjecture to more than three integers.
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Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers n as maxima of the above function.
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The smallest group containing the natural numbers is the integers. If 1 is defined as , then . That is, is simply the successor of .
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Radix sort can be applied to data that can be sorted lexicographically, be they integers, words, punch cards, playing cards, or the mail.
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He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. He is also known as "Ganitanand".
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In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend.
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This proof builds on Lagrange's result that if p=4n+1 is a prime number, then there must be an integer m such that m^2 + 1 is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since does not divide either of the Gaussian integers m + i and m-i (as it does not divide their imaginary parts), but it does divide their product m^2 + 1, it follows that p cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and p^2 = N(p), there can only be up to two factors of p), so it must be of the form p = (x+yi)(x-yi) for some integers x and y. This immediately yields that p = x^2 + y^2.
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Every element of Sp4 can be written uniquely in one of the following forms: :: [c] (ac)m [a] :: [d] (bd)n [b] :: [c] (ac)m ad (bd)n [b] where m and n are non-negative integers and terms in square brackets may be omitted as long as the remaining product is not empty. The forms of these elements imply that Sp4 has a partition Sp4 = A ∪ B ∪ C ∪ D ∪ E where :: A = { a(ca)n, (bd)n+1, a(ca)md(bd)n : m, n non-negative integers } :: B = { (ac)n+1, b(db)n, a(ca)m(db) n+1 : m, n non-negative integers } :: C = { c(ac)m, (db)n+1, (ca)m+1(db)n+1 : m, n non-negative integers } :: D = { d(bd)n, (ca)m+1(db)n+1d : m, n non-negative integers } :: E = { (ca)m : m positive integer } The sets A, B, C, D are bicyclic semigroups, E is an infinite cyclic semigroup and the subsemigroup D ∪ E is a nonregular semigroup.
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The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.
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An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers of an algebraic number field . A prime element is an element of such that if divides a product , then it divides one of the factors or . This property is closely related to primality in the integers, because any positive integer satisfying this property is either or a prime number.
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This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (). In other words, the average gap between consecutive prime numbers among the first integers is roughly .
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In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments. The Euclidean algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers, but differs in two respects. As before, the task at each step is to identify a quotient and a remainder such that :r_k = r_{k-2} - q_k r_{k-1}, where , where , and where every remainder is strictly smaller than its predecessor: . The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers.
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The extended Euclidean algorithm for the greatest common divisor of two integers and is certifying: it outputs three integers (the divisor), , and , such that . This equation can only be true of multiples of the greatest common divisor, so testing that is the greatest common divisor may be performed by checking that divides both and and that this equation is correct.
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This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in but it is not in , the ring of Gaussian integers, since and 2 does not divide any factor on the right.
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A module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions. A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group.
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Gödel has a module system, and it supports arbitrary precision integers, arbitrary precision rationals, and also floating-point numbers. It can solve constraints over finite domains of integers and also linear rational constraints. It supports processing of finite sets. It also has a flexible computation rule and a pruning operator which generalises the commit of the concurrent logic programming languages.
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A Pythagorean triple can be generated using any two positive integers by the following procedures using generalized Fibonacci sequences. For initial positive integers hn and hn+1, if and , then :(2h_{n+1} h_{n+2}, h_nh_{n+3}, 2h_{n+1}h_{n+2}+h_n ^2) is a Pythagorean triple.Horadam, A. F., "Fibonacci number triples", American Mathematical Monthly 68, 1961, 751-753.
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The set of finite and cofinite sets of integers, where a cofinite set is one omitting only finitely many integers. This is clearly closed under complement, and is closed under union because the union of a cofinite set with any set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged. Example 4.
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The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on sums of powers of integers, and the Goldbach conjecture on sums of primes. It was initiated by Hua LuogengL. K. Hua: Some results in additive prime number theory, Quart.
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Chemical formulae most often use integers for each element. However, there is a class of compounds, called non-stoichiometric compounds, that cannot be represented by small integers. Such a formula might be written using decimal fractions, as in Fe0.95O, or it might include a variable part represented by a letter, as in Fe1–xO, where x is normally much less than 1.
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In number theory, an aurifeuillean factorization, or aurifeuillian factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.
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Signed 32- and 64-bit integers will only hold at most 6 or 13 base-36 digits, respectively (that many base-36 digits can overflow the 32- and 64-bit integers). For example, the 64-bit signed integer maximum value of "9223372036854775807" is "" in base-36. Similarly, the 32-bit signed integer maximum value of "2147483647" is "" in base-36.
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In mathematics, negaFibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end. The Fibonacci code is closely related to negaFibonacci representation, a positional numeral system sometimes used by mathematicians.
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In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p, or decreases by one with probability 1-p, so the index set of this random walk is the natural numbers, while its state space is the integers. If the p=0.5, this random walk is called a symmetric random walk.
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A lucky labelling of a graph G is an assignment of positive integers to the vertices of G such that if S(v) denotes the sum of the labels on the neighbours of v, then S is a vertex coloring of G. The "lucky number" of G is the least k such that G has a lucky labelling with the integers }.
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A Hasse diagram of a portion of the lattice of ideals of the integers \Z. The purple nodes indicate prime ideals. The purple and green nodes are semiprime ideals, and the purple and blue nodes are primary ideals. In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
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By using Gödel numberings, the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers. If integers are encoded by Gödel numbers in a standard way, the arithmetic operations including addition, subtraction, and multiplication are all primitive recursive. Similarly, if the rationals are represented by Gödel numbers then the field operations are all primitive recursive.
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But if one takes such that , by the pigeonhole principle there must be } such that and are in the same integer subdivision of size (there are only such subdivisions between consecutive integers). In particular, one can find such that is in , and is in , for some integers and in }. One can then easily verify that is in . This implies that , where or .
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For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of . For example, over the integers modulo , the derivative of the polynomial is the polynomial .
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The smallest square that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 11 × 11 square, and the tiling uses five rectangles. The smallest rectangle that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, and the tiling uses five rectangles.
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However, arithmetic right shifts are major traps for the unwary, specifically in treating rounding of negative integers. For example, in the usual two's complement representation of negative integers, −1 is represented as all 1's. For an 8-bit signed integer this is 1111 1111\. An arithmetic right-shift by 1 (or 2, 3, …, 7) yields 1111 1111 again, which is still −1.
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Most used languages codes in RTF are integers slightly over 1024. 1024×768 pixels and 1280×1024 pixels are common standards of display resolution.
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In base 10, raising the digits of 175 to powers of successive integers equals itself: 135, 518, 598, and 1306 also have this property.
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Wylbur has the ability to convert line numbers between edit and IBM data sets, either as scaled integers or with an explicit decimal point.
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The concrete types of some programming languages, such as integers and strings, depend on practical issues of computer architecture, compiler implementation, and language design.
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In mathematics, a Parovicenko space is a space similar to the space of non- isolated points of the Stone–Čech compactification of the integers.
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Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
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The sequence of positive integers which have only one representation as a sum of four squares (up to order) is: :1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896 ... . These integers consist of the seven odd numbers 1, 3, 5, 7, 11, 15, 23 and all numbers of the form 2(4^k),6(4^k) or 14(4^k). The sequence of positive integers which cannot be represented as a sum of four non-zero squares is: :1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896 ... . These integers consist of the eight odd numbers 1, 3, 5, 9, 11, 17, 29, 41 and all numbers of the form 2(4^k),6(4^k) or 14(4^k).
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In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
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Example: If \Phi(X) is the signature of the oriented manifold X, then \Phi is a genus from oriented manifolds to the ring of integers.
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Magazine 74 (2001), 222–227. In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.
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Write Rm,n for the m+n dimensional vector space Rm+n with the inner product of (a1,...,am+n) and (b1,...,bm+n) given by :a1b1+...+ambm − am+1bm+1 − ... − am+nbm+n. The lattice II25,1 is given by all vectors (a1,...,a26) in R25,1 such that either all the ai are integers or they are all integers plus 1/2, and their sum is even.
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In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by . They appeared as the semigroups of values of Arf rings. A subset of the integers forms a monoid if it includes zero, and if every two elements in the subset have a sum that also belongs to the subset. In this case, it is called a "numerical semigroup".
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These two functions and form an inverse pair, and the partition generated via the Lambek–Moser theorem from this pair is just the partition of the positive integers into even and odd numbers. Lambek and Moser discuss formulas involving the prime- counting function for the functions and arising in this way from the partition of the positive integers into prime numbers and composite numbers.
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If A is a sequence of integers greater than one, and if M(A) denotes the set of positive integer multiples of members of A, then A is a Behrend sequence if M(A) has natural density one. This means that the proportion of the integers from 1 to n that belong to M(A) converges, in the limit of large n, to one.
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The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition and negation, obtained by omitting negation. By contrast, the monoid (N,+,0) of natural numbers under addition is not the reduct of any group. Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.
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The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not). The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.
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The gamma function, in blue, plotted along with in green. Notice the intersection at positive integers, both are valid analytic continuations of the factorials to the non-integers A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, Extract of page 28 Expression G.2 on page 293 :f(1) = 1, :f(x+1) = x f(x), for any positive real number . But this would allow for multiplication by any periodic analytic function which evaluates to 1 on the positive integers, such as . One of several ways to finally resolve the ambiguity comes from the Bohr–Mollerup theorem.
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In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted \gcd (x,y). For example, the gcd of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include greatest common factor ('gcf), etc...Some authors present ' as synonymous with greatest common divisor.
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The integers and the polynomials over a field share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a unit, ±1 in the case of integers) and a product of irreducible elements (prime numbers, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. Integral domains which share this property are called unique factorization domains (UFD). Greatest common divisors exist in UFDs, and conversely, every integral domain in which greatest common divisors exist is an UFD. Every principal ideal domain is an UFD.
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A set of integers S = {a1, a2, .... an} can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true.
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Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are 7 ways to do this, giving 7 maximal orders, which are all equivalent under cyclic permutations of the 7 coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were 8 maximal orders rather than 7, but as pointed out they are not closed under multiplication; this mistake occurs in several published papers.) The Kirmse integers and the 7 maximal orders are all isometric to the E8 lattice rescaled by a factor of 1/.
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Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.
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Hilbert formulated the problem as follows: > Given a Diophantine equation with any number of unknown quantities and with > rational integral numerical coefficients: To devise a process according to > which it can be determined in a finite number of operations whether the > equation is solvable in rational integers. The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integer" simply refers to the integers, positive, negative or zero: 0, ±1, ±2, ... . So Hilbert was asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers.
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The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3 In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The integer n (where n is the number of integers on a side) is the order of the magic square and the constant sum is called the magic constant. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be normal. Some authors take magic square to mean normal magic square.
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A magic triangle (also called a perimeter magic triangle) is an arrangement of the integers from 1 to on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. Unlike magic squares, there are different magic sums for magic triangles of the same order. Any magic triangle has a complementary triangle obtained by replacing each integer in the triangle with . This means that only half of the triangles for any order need to be checked for the magic property.
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Hilbert proved that, for every integer k > 1, every non-negative integer is the sum of a bounded number of k-th powers. In general, a set A of nonnegative integers is called a basis of order h if hA contains all positive integers, and it is called an asymptotic basis if hA contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set A is called a minimal asymptotic basis of order h if A is an asymptotic basis of order h but no proper subset of A is an asymptotic basis of order h.
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A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if is the product of the moduli, there is, in an interval of length , exactly one integer having any given set of modular values. The arithmetic of a residue numeral system is also called multi-modular arithmetic. Multi- modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken into account.
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Knuth's up-arrow notation, Conway chained arrow notation, and Bowers's operators : Notations that allow the concise representation of some extremely large integers such as Graham's number.
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Most noticeably, they are not all integers anymore, immediately indicating that an error was introduced in the storage, due to a poor choice of scaling factor.
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The programmer must select the appropriate intrinsic for the data types in use, e.g., "`_mm_add_epi16(x,y)`" for adding two vectors containing eight 16-bit integers.
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This proof was simplified by Gauss in his Disquisitiones Arithmeticae (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers.
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Different procedures for computing the limit of this fraction yield wildly different answers. One way to illustrate how different regularization methods produce different answers is to calculate the limit of the fraction of sets of positive integers that are even. Suppose the integers are ordered the usual way, : 1, 2, 3, 4, 5, 6, 7, 8, ... () At a cutoff of "the first five elements of the list", the fraction is 2/5; at a cutoff of "the first six elements" the fraction is 1/2; the limit of the fraction, as the subset grows, converges to 1/2. However, if the integers are ordered such that any two consecutive odd numbers are separated by two even numbers, : 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, ... () the limit of the fraction of integers that are even converges to 2/3 rather than 1/2.
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Sun-tzu's original formulation: In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime. The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the Sun-tzu Suan- ching in the 3rd century AD. The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any commutative ring, with a formulation involving ideals.
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In general the ℓ-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the ℓ-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds. For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first ℓ-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the ℓ-adic integers, provided ℓ is not the characteristic of the field concerned, and is dual to its Tate module.
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Otherwise, there are integers a and b, where n = ab, and . By the induction hypothesis, and are products of primes. But then is a product of primes.
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As with the natural numbers, the integers may also have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.
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The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are not powers of two.
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It has been said that Speidell's logarithms were to the base , but this is not entirely true due to complications with the values being expressed as integers.
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External arguments of Misiurewicz points, measured in turns are : where: a and b are positive integers and b is odd, subscript number shows base of numeral system.
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Thus, if BCF2 is a right-angle triangle with integral sides, the separation of the foci, linear eccentricity, minor axis and major axis are all also integers.
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In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of modular integers, see Root of unity modulo n.
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The integer unit executes most instructions with a one cycle latency and throughput except for multiply and divide. 32-bit multiplies have a five-cycle latency and a four-cycle throughput. 64-bit multiplies have an extra four cycles of latency and half the throughput. Divides have a 36-cycle latency and throughput for 32-bit integers, and for 64-bit integers, they are increased to 68 cycles.
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A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects.
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An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen (numbers). A field is a commutative ring where 0 ot = 1 and every non-zero element a is invertible; i.e.
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All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.
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MAA, p. 34. The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the first n odd integers is n^2, and the solution to the congruum problem.McClenon, R. B., "Leonardo of Pisa and his Liber Quadratorum", American Mathematical Monthly, Vol. 26, No. 1, January 1919, pp. 1–8.
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One may consider the ring of integers mod n, where n is squarefree. By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the factor's only idempotents will be 0 and 1. That is, each factor has two idempotents. So if there are m factors, there will be 2m idempotents.
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The theory of elliptic surfaces is analogous to the theory of proper regular models of elliptic curves over discrete valuation rings (e.g., the ring of p-adic integers) and Dedekind domains (e.g., the ring of integers of a number field). In finite characteristic 2 and 3 one can also get quasi-elliptic surfaces, whose fibers may almost all be rational curves with a single node, which are "degenerate elliptic curves".
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Let be positive integers. If :q_1 + q_2 + \cdots + q_n - n + 1 objects are distributed into boxes, then either the first box contains at least objects, or the second box contains at least objects, ..., or the th box contains at least objects. The simple form is obtained from this by taking , which gives objects. Taking gives the more quantified version of the principle, namely: Let and be positive integers.
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The pair of integers and such that is unique, in the sense that there can't be other pair of integers that satisfies the same condition in the Euclidean division theorem. In other words, if we have another division of by , say with , then we must have that :. To prove this statement, we first start with the assumptions that : : : : Subtracting the two equations yields :. So is a divisor of .
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In combinatorial number theory, the Lambek–Moser theorem is a generalization of Beatty's theorem that defines a partition of the positive integers into two subsets from any monotonic integer-valued function. Conversely, any partition of the positive integers into two subsets may be defined from a monotonic function in this way. The theorem was discovered by Leo Moser and Joachim Lambek. provides a visual proof of the result.
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This program reads in integers until a negative integer is read. It then outputs the sum of all the positive integers. Start: read // read n -> acc jmpn Done // jump to Done if acc < 0. add sum // add sum to the acc store sum // store the new sum jump Start // go back & read in next number Done: load sum // load the final sum write // write the final sum stop // stop sum: .
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The problem asks if it is possible to color each of the positive integers either red or blue, so that no Pythagorean triple of integers a, b, c, satisfying a^2+b^2=c^2 are all the same color. For example, in the Pythagorean triple 3, 4 and 5 (3^2+4^2=5^2), if 3 and 4 are colored red, then 5 must be colored blue.
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The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance to have expected running time L_n\left[\tfrac12,1+o(1)\right] by replacing the GRH assumption with the use of multipliers. The algorithm uses the class group of positive binary quadratic forms of discriminant Δ denoted by GΔ. GΔ is the set of triples of integers (a, b, c) in which those integers are relative prime.
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Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 1018, but still no general proof has been found.. In press.
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Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers.
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It is used in a special case of integer programming, in which all the decision variables are integers. It can assume the values either as zero or one.
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In fact is solution of the equation :x^2-2ax+a^2+b^2, and this equation has integer coefficients if and only if and are both integers.
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Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.
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According to an anecdote of uncertain reliability, young Carl Friedrich Gauss in primary school reinvented this method to compute the sum of the integers from 1 through 100.
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The real numbers ω1, ω2, ... , ωn are said to be rationally dependent if there exist integers k1, k2, ... , kn, not all of which are zero, such that : k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0. If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: ω1, ω2, ... , ωn are rationally independent if the only n-tuple of integers k1, k2, ... , kn such that : k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0 is the trivial solution in which every ki is zero. The real numbers form a vector space over the rational numbers, and this is equivalent to the usual definition of linear independence in this vector space.
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The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring. An example of an exponential ring with a nontrivial exponential function is the ring of integers Z equipped with the function E which takes the value +1 at even integers and −1 at odd integers, i.e., the function n \mapsto (-1)^n. This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions.
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There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara II. The problem of representing integers by sums of two squares was considered in the 3rd century by Diophantus.
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Büchi arithmetic in base a and b define the same sets if and only if a and b are multiplicatively dependent. Let a and b be multiplicatively dependent integers, that is, there exists n,m>1 such that a^n=b^m. The integers c such that the length of its expansion in base a is at most m are exactly the integers such that the length of their expansion in base b is at most n. It implies that computing the base b expansion of a number, given its base a expansion, can be done by transforming consecutive sequences of m base a digits into consecutive sequence of n base b digits.
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MMX enhancement (1993–1999) Intel Pentium MMX microarchitecture Pentium MMX 166 MHz without cover The P55C (or 80503) was developed by Intel's Research & Development Center in Haifa, Israel. It was sold as Pentium with MMX Technology (usually just called Pentium MMX); although it was based on the P5 core, it featured a new set of 57 "MMX" instructions intended to improve performance on multimedia tasks, such as encoding and decoding digital media data. The Pentium MMX line was introduced on October 22, 1996, and released in January 1997. The new instructions worked on new data types: 64-bit packed vectors of either eight 8-bit integers, four 16-bit integers, two 32-bit integers, or one 64-bit integer.
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For example, integers are an ADT, defined as the values ..., −2, −1, 0, 1, 2, ..., and by the operations of addition, subtraction, multiplication, and division, together with greater than, less than, etc., which behave according to familiar mathematics (with care for integer division), independently of how the integers are represented by the computer. Explicitly, "behavior" includes obeying various axioms (associativity and commutativity of addition, etc.), and preconditions on operations (cannot divide by zero). Typically integers are represented in a data structure as binary numbers, most often as two's complement, but might be binary-coded decimal or in ones' complement, but the user is abstracted from the concrete choice of representation, and can simply use the data as data types.
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Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non- negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).
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A magic graph is a graph whose edges are labelled by positive integers, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. If the integers are the first q positive integers, where q is the number of edges, the graph and the labelling are called supermagic. A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant.
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The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where A, B, C, x, y, and z are non-zero integers with x, y, z ≥ 3, then A, B, and C have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in non-zero integers and pairwise coprime integers A, B, C if x, y, z ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's last theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.
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It is also 17-gonal. 613 is the sum of squares of two consecutive integers, 17 and 18, and is also a lucky number and thus a lucky prime.
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OpenCL has a built-in intrinsic for multiplication (`mul24()`) with two 24-bit integers, returning a 32-bit result. It is typically much faster than a 32-bit multiplication.
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A similar theorem was given by Kazimierz Kuratowski for Polish spaces, stating that they are isomorphic, as Borel spaces, to either the reals, the integers, or a finite set.
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The sampling rate and the number of bits used to represent the integers combine to determine how close such an approximation to the analog signal a digitization will be.
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This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (Richardson's theorem).
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308 = 22 × 7 × 11\. 308 is a nontotient, totient sum of the first 31 integers, Harshad number, heptagonal pyramidal number, and the sum of two consecutive primes (151 + 157).
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324 = 22 × 34 = 182. 324 is the sum of four consecutive primes (73 + 79 + 83 + 89), totient sum of the first 32 integers, untouchable number, and a Harshad number.
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If both operands and the desired result all have the same scaling factor, then the quotient of the two integers must be explicitly multiplied by that common scaling factor.
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This sequence started in 1983 with the publication of Galois module structure of algebraic integers by Albrecht Fröhlich. As of February 2008, the editor-in-chief is R. Remmert.
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In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory, Teichmüller representatives are sometimes used as digits.
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A preordered vector space (X, ≤) with an order unit u is Archimedean preordered if and only if n x ≤ u for all non-negative integers n implies x ≤ 0\.
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Perhaps most importantly, the use of 32-bit signed integers for addressing limits HDF4 files to a maximum of 2 GB, which is unacceptable in many modern scientific applications.
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The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. All integers between zero and are -harshad numbers.
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This section constructs the ring of integers, the fields of rational and real numbers, and "vector-families", which are related to what are now called torsors over abelian groups.
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Reduction of summands is an algorithm for fast binary multiplication of non- signed binary integers. It is performed in three steps: production of summands, reduction of summands, and summation.
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In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by . The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic groups over fields of other characteristics or over the integers. showed that Demazure's conjecture (for classical groups) follows from their work on standard monomial theory, and Peter Littelmann extended this to all reductive algebraic groups.
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In the Standard Model, vector (spin-1) bosons (gluons, photons, and the W and Z bosons) mediate forces, whereas the Higgs boson (spin-0) is responsible for the intrinsic mass of particles. Bosons differ from fermions in the fact that multiple bosons can occupy the same quantum state (Pauli exclusion principle). Also, bosons can be either elementary, like photons, or a combination, like mesons. The spin of bosons are integers instead of half integers.
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This design was chosen to save die area. The multiplier and divider are not pipelined and have significant latencies: multiplies have a 10- or 20-cycle latency for 32-bit or 64-bit integers, respectively; whereas divides have a 69- or 133-cycle latency for 32-bit or 64-bit integers, respectively. Most instructions have a single cycle latency. The ALU adder is also used for calculating virtual addresses for loads, stores and branches.
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The even integers from 4 to 28 as sums of two primes: Even integers correspond to horizontal lines. For each prime, there are two oblique lines, one red and one blue. The sums of two primes are the intersections of one red and one blue line, marked by a circle. Thus the circles on a given horizontal line give all partitions of the corresponding even integer into the sum of two primes.
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Church's paper An Unsolvable Problem of Elementary Number Theory (1936) proved that the Entscheidungsproblem was undecidable within the λ-calculus and Gödel-Herbrand's general recursion; moreover Church cites two theorems of Kleene's that proved that the functions defined in the λ-calculus are identical to the functions defined by general recursion: :"Theorem XVI. Every recursive function of positive integers is λ-definable.16 :"Theorem XVII. Every λ-definable function of positive integers is recursive.
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The orbit of a rational map may contain infinitely many integers. For example, if is a polynomial with integer coefficients and if is an integer, then it is clear that the entire orbit consists of integers. Similarly, if is a rational map and some iterate is a polynomial with integer coefficients, then every -th entry in the orbit is an integer. An example of this phenomenon is the map , whose second iterate is a polynomial.
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This second way of specifying a domain allows for domains that are not composed of integers, such as `X::[george,mary,john]`. If the domain of a variable is not specified, it is assumed to be the set of integers representable in the language. A group of variables can be given the same domain using a declaration like `[X,Y,Z]::[1..5]`. The domain of a variable may be reduced during execution.
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In mathematics, Legendre's equation is the Diophantine equation :ax^2+by^2+cz^2=0. The equation is named for Adrien Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers, not all positive or all negative .
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As with unsigned graphs, there is a notion of signed graph coloring. Where a coloring of a graph is a mapping from the vertex set to the natural numbers, a coloring of a signed graph is a mapping from the vertex set to the integers. The constraints on proper colorings come from the edges of the signed graph. The integers assigned to two vertices must be distinct if they are connected by a positive edge.
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The fastest known algorithms for the multiplication of very large integers use the polynomial multiplication method outlined above. Integers can be treated as the value of a polynomial evaluated specifically at the number base, with the coefficients of the polynomial corresponding to the digits in that base (ex. 123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0). After polynomial multiplication, a relatively low- complexity carry-propagation step completes the multiplication.
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In LFE, the list data type is written with its elements separated by whitespace, and surrounded by parentheses. For example, is a list whose elements are the integers and , and the atom . These values are implicitly typed: they are respectively two integers and a Lisp-specific data type called a symbolic atom, and need not be declared as such. As seen in the example above, LFE expressions are written as lists, using prefix notation.
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The Havel–Hakimi algorithm is an algorithm in graph theory solving the graph realization problem. That is, it answers the following question: Given a finite list of nonnegative integers, is there a simple graph such that its degree sequence is exactly this list? The degree sequence is a list of numbers that for each vertex of the graph states how many neighbors it has. For a positive answer, the list of integers is called graphic.
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Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36, hence 36 is a semiperfect number. This number is the sum of a twin prime pair (17 + 19), the sum of the cubes of the first three positive integers, and also the product of the squares of the first three positive integers. 36 is the number of degrees in the interior angle of each tip of a regular pentagram.
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A magic polygon also called a perimeter magic polygon is a polygon with an integers on its sides that all add up to a magic sum. It is where positive integers (from 1 to N) on a k-sided polygon add up to a constant, or magic sum. Magic polygons are the generalization of other magic shapes such as magic triangles. This displays order 3 magic triangles, a type of magic polygon.
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StatView is a statistics application originally released for Apple Macintosh computers in 1985. StatView was one of the first statistics applications to have a graphical user interface, capitalizing on the Macintosh's. A user saw a spreadsheet of his or her data, comprising columns that could be integers, long integers, real numbers, strings, or categories, and rows that were usually cases (such as individual people for psychology data). Columns had informative headings; rows were numbered.
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In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers.
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The value of an item with an integral type is the mathematical integer that it corresponds to. Integral types may be unsigned (capable of representing only non-negative integers) or signed (capable of representing negative integers as well). An integer value is typically specified in the source code of a program as a sequence of digits optionally prefixed with + or −. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8).
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Length is encoded following the same convention as integers, thus including its own type. For example, the string `hello` is encoded as `S`,`U`,0x05,`h`,`e`,`l`,`l`,`o`.
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Set of real numbers (R), which include the rationals (Q), which include the integers (Z), which include the natural numbers (N). The real numbers also include the irrationals (R\Q).
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Order: 214 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 13 ⋅ 29 = 145926144000 Schur multiplier: Order 2. Outer automorphism group: Trivial. Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers.
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Many of the implementation methods were borrowed from Maclisp: bibop memory organization (BIg Bag Of Pages), small integers represented uniquely by pointers to fixed values in fields, and fast arithmetic.
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Despite this theoretical limit, in practice, theorem provers can solve many hard problems, even in models that are not fully described by any first order theory (such as the integers).
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Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.
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Then has sum 0, and so must . But since can only contain positive integers, it must be empty too. To conclude: ∅ which implies , proving that each Zeckendorf representation is unique.
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The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complement, as well as the arithmetic operations addition and multiplication.
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In a computer-aided proof, Krasikov and Lagarias showed that the number of integers in the interval that eventually reach 1 is at least equal to for all sufficiently large .
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The square root version, the EC-132, effectively subtracted consecutive odd integers, each decrement requiring two consecutive subtractions. After the first, the minuend was incremented by one before the second subtraction.
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Similar to a Pythagorean triple, an Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees.
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OEIS also lists the number of pairs of this type where the larger of the two integers in the pair is square or triangular , as both types of pair arise frequently.
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A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
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90,000 (ninety thousand) is the natural number following 89,999 and preceding 90,001. It is the sum of the cubes of the first 24 positive integers, and is the square of 300.
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In a still another context, the terminology is used to describe a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel.
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What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers. A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers (i.e.
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The local store does not operate like a conventional CPU cache since it is neither transparent to software nor does it contain hardware structures that predict which data to load. The SPEs contain a 128-bit, 128-entry register file and measures 14.5 mm2 on a 90 nm process. An SPE can operate on sixteen 8-bit integers, eight 16-bit integers, four 32-bit integers, or four single-precision floating-point numbers in a single clock cycle, as well as a memory operation. Note that the SPU cannot directly access system memory; the 64-bit virtual memory addresses formed by the SPU must be passed from the SPU to the SPE memory flow controller (MFC) to set up a DMA operation within the system address space.
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Euclidean division of polynomials, which is used in Euclid's algorithm for computing GCDs, is very similar to Euclidean division of integers. Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which satisfy :a=bq+r and :\deg(r)<\deg(b), where "deg(...)" denotes the degree and the degree of the zero polynomial is defined as being negative. Moreover, q and r are uniquely defined by these relations. The difference from Euclidean division of the integers is that, for the integers, the degree is replaced by the absolute value, and that to have uniqueness one has to suppose that r is non-negative.
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In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering. Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
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Secondly, this method uses negative numbers as necessary, even when multiplying two positive integers, in order to quicken the rate of multiplication via subtraction. This means two positive integers can be multiplied together to get negative intermediate steps, yet still the correct positive answer in the end. These negative numbers are actually automatically derived from the multiplication steps themselves and are thus unique to a particular problem. Again, such negative intermediate steps are designed to help hasten the mental math. Finally, another unique aspect of using this method is that the user is able to choose one of several different “routes of multiplication” to the specific multiplication problem at hand based on their subjective preferences or strengths and weaknesses with particular integers.
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Lagrange's four- square theorem is a special case of the Fermat polygonal number theorem and Waring's problem. Another possible generalization is the following problem: Given natural numbers a,b,c,d, can we solve :n=ax_1^2+bx_2^2+cx_3^2+dx_4^2 for all positive integers n in integers x_1,x_2,x_3,x_4? The case a=b=c=d=1 is answered in the positive by Lagrange's four-square theorem. The general solution was given by Ramanujan.. He proved that if we assume, without loss of generality, that a\leq b\leq c\leq d then there are exactly 54 possible choices for a,b,c,d such that the problem is solvable in integers x_1,x_2,x_3,x_4 for all n.
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The existence of such a "bad universal sequence" came as a surprise. Bellow showed that every lacunary sequence of integers is in fact a "bad universal sequence" in L_1. Thus lacunary sequences are ‘canonical’ examples of "bad universal sequences". Later she was able to show that from the point of view of the pointwise ergodic theorem, a sequence of positive integers may be "good universal" in L_p, but "bad universal" in L_q, for all 1\le q < p.
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The conjecture was posed by Émile Lemoine in 1895, but was erroneously attributed by MathWorld to Hyman Levy who pondered it in the 1960s. A similar conjecture by Sun in 2008 states that all odd integers greater than 3 can be represented as the sum of a prime number and the product of two consecutive positive integers ( p+x(x+1) ).Sun, Zhi-Wei. "On sums of primes and triangular numbers." arXiv preprint arXiv:0803.3737 (2008).
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Palm OS uses both signed integers with the 1970 epoch, as well as unsigned integers with the 1904 epoch, for different system functions, such as for system clock, and file dates (see PDB format). While this should result in Palm OS being susceptible to the 2038 problem, Palm OS also uses a 7-bit field for storing the year value, with a different epoch counting from 1904, resulting in a maximum year of (1904+127) 2031.
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Sorting algorithms in general sort a list of objects according to some ordering scheme. In contrast to comparison-based sorting algorithms, such as quicksort, American flag sort can only sort integers (or objects that can be interpreted as integers). In-place sorting algorithms, including American flag sort, run without allocating a significant amount of memory beyond that used by the original array. This is a significant advantage, both in memory savings and in time saved copying the array.
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Each such reading is called a sample and may be considered to have infinite precision at this stage; ;Quantization: Samples are rounded to a fixed set of numbers (such as integers), a process known as quantization. In general, these can occur at the same time, though they are conceptually distinct. A series of digital integers can be transformed into an analog output that approximates the original analog signal. Such a transformation is called a DA conversion.
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Thus the computation of polynomial GCD's is essentially the same problem over F[X] and over R[X]. For univariate polynomials over the rational numbers, one may think that Euclid's algorithm is a convenient method for computing the GCD. However, it involves simplifying a large number of fractions of integers, and the resulting algorithm is not efficient. For this reason, methods have been designed to modify Euclid's algorithm for working only with polynomials over the integers.
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Integer precision (bit-size) must be kept during vector instruction execution. The correct vector instruction must be chosen based on the size and behavior of the internal integers. Also, with mixed integer types, extra care must be taken to promote/demote them correctly without losing precision. Special care must be taken with sign extension (because multiple integers are packed inside the same register) and during shift operations, or operations with carry bits that would otherwise be taken into account.
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In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end. The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1s.
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OpenLisp uses tagged architecture (4 bits tag on 32-bit, 5 bits tag on 64-bit) for fast type checking (small integer, float, symbol, cons, string, vector). Small integers (28 bits on 32-bit, 59 bits on 64-bit) are unboxed, large (32/64-bit) integers are boxed. As required by ISLISP, arbitrary-precision arithmetic (bignums) are also implemented. Characters (hence strings) are either 8-bit (ANSI, EBCDIC) or 16/32-bit if Unicode support is enabled.
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The expression is an effective initial guess for computing the square root of a 32-bit integer using Newton's method. CLZ can efficiently implement null suppression, a fast data compression technique that encodes an integer as the number of leading zero bytes together with the nonzero bytes. It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers. The log base 2 can be used to anticipate whether a multiplication will overflow, since .
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Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups.
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The first thousand values of . The points on the top line represent when is a prime number, which is In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1.
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The signal may be real valued or complex-valued, defined on a continuous set (for example, the real numbers) or a discrete set (for example, the integers or a finite subset of integers). The Zak transform is a generalization of the discrete Fourier transform. The Zak transform had been discovered by several people in different fields and was called by different names. It was called the "Gel'fand mapping" because I.M. Gel'fand introduced it in his work on eigenfunction expansions.
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The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called nontrivial zeros.
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We now fix a monoid (M,0,+) which is a subset of the real line. This monoid is traditionally the set of integers, rationals, reals, or their subset of non-negative numbers.
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Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
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In applications s is usually on the critical line, and the positive integers M and N are chosen to be about . found good bounds for the error of the Riemann–Siegel formula.
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This is related by Nicomachus's theorem to the fact that 100 also equals the square of the sum of the first four integers: . 26 \+ 62 = 100, thus 100 is a Leyland number.
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XCFiles is intended to be portable to any 32-bit platform which meets certain requirements (such as supporting semaphores and unsigned 64-bit integers).exFiles User's Manual (v. 1.04), pp. 67, 72.
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There are several common algorithms for hashing integers. The method giving the best distribution is data-dependent. One of the simplest and most common methods in practice is the modulo division method.
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This process is similar to the method taught to primary schoolchildren for conducting long multiplication on base-10 integers, but has been modified here for application to a base-2 (binary) numeral system.
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The spectrum of a McKay–Miller–Širáň graph has at most five distinct eigenvalues. In some of these graphs, all of these values are integers, so that the graph is an integral graph.
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In the context of integers, subtraction of one also plays a special role: for any integer a, the integer is the largest integer less than a, also known as the predecessor of a.
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It is a nontotient since there is no integer with exactly 122 coprimes below it. Nor is there an integer with exactly 122 integers with common factors below it, making 122 a noncototient.
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If and are integers, rationals, or real numbers, then implies or . Consider . Then, substituting for and for , we learn or . Then we can substitute again, letting and , to show that if then or .
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The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120.
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In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by . They are functions whose zeroes and poles are confined to the cusps (images of infinity).
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The mathematics section of the test focuses on several math categories. The math problems include: Integers, Fractions, Decimals, Percents, Algebra, Plane Geometry, Polygons, Circles, Measurements, Graphs and Tables, Word Problems, Sequences, and Analogies.
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Every linearly ordered field contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of , which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in . The following are equivalent characterizations of Archimedean fields in terms of these substructures. 1\. The natural numbers are cofinal in .
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In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem. Another example are Gaussian integers, that is, numbers of the form , where and are integers, which can be used to classify sums of squares.
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The second column of the table contains only integers in the first quadrant, which means the real part x is positive and the imaginary part y is non-negative. The table might have been further reduced to the integers in the first octant of the complex plane using the symmetry . The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry , for example, could also be written as .
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The JVM operates on primitive values (integers and floating-point numbers) and references. The JVM is fundamentally a 32-bit machine. `long` and `double` types, which are 64-bits, are supported natively, but consume two units of storage in a frame's local variables or operand stack, since each unit is 32 bits. `boolean`, `byte`, `short`, and `char` types are all sign-extended (except `char` which is zero-extended) and operated on as 32-bit integers, the same as `int` types.
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The Bernoulli numbers can be expressed in terms of the Riemann zeta function as for integers provided for the expression is understood as the limiting value and the convention is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that is a prime number if and only if is congruent to −1 modulo .
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The R4600 was a simple design; it was a scalar processor, issuing up to one instruction per cycle to its integer pipeline or floating-point unit (FPU). Most integer instructions have a single cycle latency and throughput, except for multiplies and divides. Multiplies, 32-bit and 64-bit, have an eight-cycle latency and six-cycle throughput. Divides have a 32-cycle latency and throughput for 32-bit integers and a 61-cycle latency and throughput for 64-bit integers.
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The goal of TWINKLE is to implement the sieving step of the Number Field Sieve algorithm, which is the fastest known algorithm for factoring large integers. The sieving step, at least for 512-bit and larger integers, is the most time consuming step of NFS. It involves testing a large set of numbers for B-'smoothness', i.e., absence of a prime factor greater than a specified bound B. What is remarkable about TWINKLE is that it is not a purely digital device.
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Their logarithms did not lead to irrational numbers, however Theon tackled this discussion head on. He acknowledged that “one can prove that” the tone of value 9/8 cannot be divided into equal parts and so it is a number in itself. Many Pythagoreans believed in the existence of irrational numbers, but did not believe in using them because they were unnatural and not positive integers. Theon also does an amazing job of relating quotients of integers and musical intervals.
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Although the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers), it may be generalized to the real numbers, and to other mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory.
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The cases and yield the Gaussian integers and Eisenstein integers, respectively. If is allowed to be any Euclidean function, then the list of possible values of for which the domain is Euclidean is not yet known. The first example of a Euclidean domain that was not norm-Euclidean (with ) was published in 1994. In 1973, Weinberger proved that a quadratic integer ring with is Euclidean if, and only if, it is a principal ideal domain, provided that the generalized Riemann hypothesis holds.
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1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, , but if units are included, is also equal to, say, among infinitely many similar "factorizations". 1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1).
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The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.
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The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of q and r. Other proofs use the well-ordering principle (i.e., the assertion that every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage of not providing directly an algorithm for solving the division (see for more).
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140 is an abundant number and a harmonic divisor number. It is the sum of the squares of the first seven integers, which makes it a square pyramidal number, and in base 10 it is divisible by the sum of its digits, which makes it a Harshad number. 140 is an odious number because it has an odd number of ones in its binary representation. The sum of Euler's totient function φ(x) over the first twenty-one integers is 140.
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Let I be an ideal of a ring R. The sets of the form , for all x in R and all positive integers n, form a base for a topology on R that makes R into a topological ring. Then for any left R-module M, the sets of the form , for all x in M and all positive integers n, form a base for a topology on M that makes M into a topological module over the topological ring R.
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Suppose that p and q are rational primes congruent to 1 mod 4 such that the Legendre symbol (p/q) is 1. Then the ideal (p) factorizes in the ring of integers of Q() as (p)=𝖕𝖕' and similarly (q)=𝖖𝖖' in the ring of integers of Q(). Write εp and εq for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that :[εp/𝖖] = [εq/𝖕] where [] is the quadratic residue symbol in a quadratic number field.
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An integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrange's four- square theorem shows that w^2+x^2+y^2+z^2 is universal. Ramanujan generalized this to aw^2+bx^2+cy^2+dz^2 and found 54 multisets that can each generate all positive integers, namely, :{1, 1, 1, d}, 1 ≤ d ≤ 7 :{1, 1, 2, d}, 2 ≤ d ≤ 14 :{1, 1, 3, d}, 3 ≤ d ≤ 6 :{1, 2, 2, d}, 2 ≤ d ≤ 7 :{1, 2, 3, d}, 3 ≤ d ≤ 10 :{1, 2, 4, d}, 4 ≤ d ≤ 14 :{1, 2, 5, d}, 6 ≤ d ≤ 10 There are also forms whose image consists of all but one of the positive integers. For example, {1,2,5,5} has 15 as the exception.
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Balanced ternary is a non-standard positional numeral system (a balanced form), used in some early computers and useful in the solution of balance puzzles. It is a ternary (base 3) number system in which the digits have the values –1, 0, and 1, in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. Balanced ternary can represent all integers without using a separate minus sign; the value of the leading non- zero digit of a number has the sign of the number itself. While binary numerals with digits 0 and 1 provide the simplest positional numeral system for natural numbers (or for positive integers if using 1 and 2 as the digits), balanced ternary provides the simplest self-contained positional numeral system for integers.
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Euler's "lucky" numbers are positive integers n such that for all integers k with , the polynomial produces a prime number. When k is equal to n, the value cannot be prime since is divisible by n. Since the polynomial can be written as , using the integers k with produces the same set of numbers as . Leonhard Euler published the polynomial which produces prime numbers for all integer values of k from 1 to 40. Only 7 lucky numbers of Euler exist, namely 1, 2, 3, 5, 11, 17 and 41 . The primes of the form k2 − k + 41 are :41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, ... .
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Connascence of meaning is when multiple components must agree on the meaning of particular values. Returning integers 0 and 1 to represent false and true, respectively, is an example of this form of connascence.
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It follows from this definition that a function f is automatically continuous at every isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.
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Specifically it states that for any integers and , the functions and have no common zeros other than the one at . The hypothesis was proved by Carl Ludwig Siegel in 1929.Watson, pp. 484–485.
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The existence of irrational numbers—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.
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Albrecht Fröhlich FRS (22 May 1916 – 8 November 2001) was a German-born British mathematician, famous for his major results and conjectures on Galois module theory in the Galois structure of rings of integers.
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In mathematics, a real closed ring is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi- algebraic functions defined over the integers.
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If all of the diagonals of a regular decagon are drawn, the resulting figure will have exactly 220 regions. It is the sum of the sums of the divisors of the first 16 positive integers.
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A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not.
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Rational points (x,y)=(a/c,b/c) on the unit circle x^2+y^2=1 correspond to Pythagorean triples, i.e. triples (a,b,c) of integers satisfying a^2+b^2=c^2.
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In the case of non-orientable manifolds, every homology class of H_n(X,\Z_2), where \Z_2 denotes the integers modulo 2, can be realized by a non-oriented manifold, f\colon M^n\to X.
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The Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It was published independently by Nagell and by Élisabeth Lutz.
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In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture and participated in the Cunningham project.
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The density is at most \Gamma(4/3)^3/6\approx 0.119. Every integer can be represented as a sum of three cubes of rational numbers (rather than as a sum of cubes of integers).
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For unsigned integers, the bitwise complement of a number is the "mirror reflection" of the number across the half-way point of the unsigned integer's range. For example, for 8-bit unsigned integers, `NOT x = 255 - x`, which can be visualized on a graph as a downward line that effectively "flips" an increasing range from 0 to 255, to a decreasing range from 255 to 0. A simple but illustrative example use is to invert a grayscale image where each pixel is stored as an unsigned integer.
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However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors.
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Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (over a trillion).
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The notion of unimodular matrix of integers must be extended by calling unimodular a matrix over an integral domain whose determinant is a unit. This means that the determinant is invertible and implies that unimodular matrices are the invertible matrices such all entries of the inverse matrix belong to the domain. To have an algorithmic solution of linear systems, a solution for a single linear equation in two unknowns is clearly required. In the case of the integers, such a solution is provided by extended Euclidean algorithm.
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However, it is worth noting that 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1. Guy and Selfridge believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are unbounded above (or diverge)).A. S. Mosunov, What do we know about aliquot sequences? , there were 898 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9190 such integers less than 1,000,000.
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From a heuristic view, we expect the probability that the ratio of the length of the gap to the natural logarithm is greater than or equal to a fixed positive number k to be ; consequently the ratio can be arbitrarily large. Indeed, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius.. In the opposite direction, the twin prime conjecture posits that for infinitely many integers n.
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In other words, the GCD is unique up to the multiplication by an invertible constant. In the case of the integers, this indetermination has been settled by choosing, as the GCD, the unique one which is positive (there is another one, which is its opposite). With this convention, the GCD of two integers is also the greatest (for the usual ordering) common divisor. However, since there is no natural total order for polynomials over an integral domain, one cannot proceed in the same way here.
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A mesh edge is defined by a pair of integers hv0,v1i, each integer corresponding to an end point of the edge. To support edge maps, the edges are stored so that v0 = min(v0,v1). A triangle component is defined by a triple of integers hv0,v1,v2i, each integer corresponding to a vertex of the triangle. To support triangle maps, the triangles are stored so that v0 = min(v0,v1,v2). Observe that hv0,v1,v2i and hv0,v2,v1i are treated as different triangles.
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These were similar to the virtual arrays in BASIC-PLUS in but more limited. An array of integers, floatingpoint, or character strings of length 1, 2, 4, 8, 16, 32, or 64 could be placed in file and accessed with a subscript. The file could actually be opened (or re- opened) with a different definition allowing integers, characters, and floating point numbers to be stored in the same file. Like BASIC-11, Multi- User BASIC provided some support for lab equipment, support for character terminals (LA30, VT100).
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To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow occurs (i.e. provided that the sum of the two integers fits in the underlying integer type). If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.
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These cannot be replaced by any finite number of axioms, that is, Presburger arithmetic is not finitely axiomatizable in first-order logic. Presburger arithmetic can be viewed as first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers. Presburger arithmetic is designed to be complete and decidable.
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There are exactly 126 positive integers that are not solutions of the equation :x=abc+abd+acd+bcd, where a, b, c, and d must themselves all be positive integers.. See OEIS:A027563 for the list of these 126 numbers. It is the fifth Granville number, and the third such not to be a perfect number. Also, it is known to be the smallest Granville number with three distinct prime factors, and perhaps the only such Granville number.J. D. Koninck, Those Fascinating Numbers, transl. author.
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A variation of the paradox uses integers instead of real-numbers, while preserving the self- referential character of the original. Consider a language (such as English) in which the arithmetical properties of integers are defined. For example, "the first natural number" defines the property of being the first natural number, one; and "divisible by exactly two natural numbers" defines the property of being a prime number. (It is clear that some properties cannot be defined explicitly, since every deductive system must start with some axioms.
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That is, if x2, y2, and z2 (for integers x, y, and z) are three square numbers that are equally spaced apart from each other, then the spacing between them, , is called a congruum. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers x, y, and z such that :y^2 - x^2 = z^2 - y^2. When this equation is satisfied, both sides of the equation equal the congruum.
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Google's Protocol Buffers "zig-zag encoding" is a system similar to sign-and-magnitude, but uses the least significant bit to represent the sign and has a single representation of zero. This allows a variable-length quantity encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers.Protocol Buffers: Signed Integers Another approach is to give each digit a sign, yielding the signed-digit representation. For instance, in 1726, John Colson advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5.
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The Leech lattice is also a 12-dimensional lattice over the Eisenstein integers. This is known as the complex Leech lattice, and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the binary Golay code is replaced with the ternary Golay code, and the Mathieu group M24 is replaced with the Mathieu group M12. The E6 lattice, E8 lattice and Coxeter-Todd lattice also have constructions as complex lattices, over either the Eisenstein or Gaussian integers.
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The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the exponents are required to be rational integers and linear independence is only assured over the rational integers,. a result sometimes referred to as Hermite's theorem.. Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882.
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In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, 6, ...} is a numerical semigroup, the set {0, 1, 3, 5, 6, ...} is not because 1 is in the set and 1 + 1 = 2 is not in the set.
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In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words.
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In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}.
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Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings.
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In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4^a(8b + 7)) are :7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... .
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The programmer is free to push arbitrary-width integers onto the stack (currently there is no implementation of floating point numbers) and can also access the heap as a permanent store for variables and data structures.
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HAL/S has native support for integers, floating point scalars, vector, matrices, booleans and strings of 8-bit characters, limited to a maximum length of 255. Structured types may be composed using a `DECLARE STRUCT` statement.
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To see that the second condition is sharp, consider the function . It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of , and indeed it is not identically zero.
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Cunningham numbers are a simple type of binomial number, they are of the form :b^n\pm1 where b and n are integers and b is not a perfect power. They are denoted C±(b, n).
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Frenicle's Methode, 1754 edition. He challenged Christiaan Huygens to solve the following system of equations in integers, :x2 \+ y2 = z2, x2 = u2 \+ v2, x − y = u − v. A solution was given by Théophile Pépin in 1880.
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To further generalize, an A-restricted composition of an integer n, for a subset A of the (nonnegative or positive) integers, is an ordered collection of one or more elements in A whose sum is n.
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The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime.
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There are no more than cosets, because . The coset is the congruence class of modulo . The subgroup is normal in , and so, can be used to form the quotient group the group of integers mod m.
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The single list is the sorted list. The merge algorithm is used repeatedly in the merge sort algorithm. An example merge sort is given in the illustration. It starts with an unsorted array of 7 integers.
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In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It is named for Trygve Nagell and Élisabeth Lutz.
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Subtraction and bitwise logical operations on 16 bits is done in 8-bit steps. Operations that have to be implemented by program code (subroutine libraries) include comparisons of signed integers as well as multiplication and division.
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The cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are cyclically permuted when multiplied by certain numbers. For example, . 102564 is the repetend of and 410256 the repetend of .
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As of 2018, the problem is still open for more than 2 colors, that is, if there exists a k-coloring (k ≥ 3) of the positive integers such that no Pythagorean triples are the same color.
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A: Math. Gen. 36 (2003), no. 24, 6651.cond-mat/0303607 In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers n\leq 72.
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Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders. Its existence is based on the following theorem: Given two univariate polynomials a(x) and b(x) (where b(x) is a non-zero polynomial) defined over a field (in particular, the reals or complex numbers), there exist two polynomials q(x) (the quotient) and r(x) (the remainder) which satisfy: :a(x) = b(x)q(x) + r(x) where :\deg(r(x)) < \deg(b(x)), where "deg(...)" denotes the degree of the polynomial (the degree of the constant polynomial whose value is always 0 can be defined to be negative, so that this degree condition will always be valid when this is the remainder). Moreover, q(x) and r(x) are uniquely determined by these relations. This differs from the Euclidean division of integers in that, for the integers, the degree condition is replaced by the bounds on the remainder r (non-negative and less than the divisor, which insures that r is unique.) The similarity between Euclidean division for integers and that for polynomials motivates the search for the most general algebraic setting in which Euclidean division is valid.
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The general constraint satisfaction problem consists in finding a list of integers , each in some range }, that satisfies some arbitrary constraint (boolean function) F. For this class of problems, the instance data P would be the integers m and n, and the predicate F. In a typical backtracking solution to this problem, one could define a partial candidate as a list of integers , for any k between 0 and n, that are to be assigned to the first k variables . The root candidate would then be the empty list (). The first and next procedures would then be function first(P, c) is k ← length(c) if k = n then return NULL else return (c[1], c[2], …, c[k], 1) function next(P, s) is k ← length(s) if s[k] = m then return NULL else return (s[1], s[2], …, s[k − 1], 1 + s[k]) Here length(c) is the number of elements in the list c. The call reject(P, c) should return true if the constraint F cannot be satisfied by any list of n integers that begins with the k elements of c.
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A Nivenmorphic number or harshadmorphic number for a given number base is an integer such that there exists some harshad number whose digit sum is , and , written in that base, terminates written in the same base. For example, 18 is a Nivenmorphic number for base 10: 16218 is a harshad number 16218 has 18 as digit sum 18 terminates 16218 Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.. In fact, for an even integer n > 1, all positive integers except n+1 are Nivenmorphic numbers for base n, and for an odd integer n > 1, all positive integers are Nivenmorphic numbers for base n. e.g. the Nivenmorphic numbers in base 12 are (all positive integers except 13). The smallest number with base 10 digit sum n and terminates n written in base 10 are: (0 if no such number exists) :1, 2, 3, 4, 5, 6, 7, 8, 9, 910, 0, 912, 11713, 6314, 915, 3616, 15317, 918, 17119, 9920, 18921, 9922, 82823, 19824, 9925, 46826, 18927, 18928, 78329, 99930, 585931, 388832, 1098933, 198934, 289835, 99936, 99937, 478838, 198939, 1999840, 2988941, 2979942, 2979943, 999944, 999945, 4698946, 4779947, 2998848, 2998849, 9999950, ...
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In Pascal, a similar end is performed by declaring a subrange of integer (a compiler may then choose to allocate a smaller amount of storage for the declared variable): type a = 1..100; b = -20..20; c = 0..100000; This subrange feature is not supported by C. A major, if subtle, difference between C and Pascal is how they promote integer operations. In Pascal, all operations on integers or integer subranges have the same effect, as if all of the operands were promoted to a full integer. In C, there are defined rules as to how to promote different types of integers, typically with the resultant type of an operation between two integers having a precision that is greater than or equal to the precisions of the operands. This can make machine code generated from C efficient on many processors.
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More generally, every system of linear Diophantine equations may be solved by computing the Smith normal form of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations over a field. Using matrix notation every system of linear Diophantine equations may be written :, where is an matrix of integers, is an column matrix of unknowns and is an column matrix of integers. The computation of the Smith normal form of provides two unimodular matrices (that is matrices that are invertible over the integers and have ±1 as determinant) and of respective dimensions and , such that the matrix : is such that is not zero for not greater than some integer , and all the other entries are zero. The system to be solved may thus be rewritten as :.
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In mathematics, an overring B of an integral domain A is a subring of the field of fractions K of A that contains A: i.e., A \subseteq B \subseteq K.. For instance, an overring of the integers is a ring in which all elements are rational numbers, such as the ring of dyadic rationals. A typical example is given by localization: if S is a multiplicatively closed subset of A, then the localization S−1A is an overring of A. The rings in which every overring is a localization are said to have the QR property; they include the Bézout domains and are a subset of the Prüfer domains.. See in particular p. 196. In particular, every overring of the ring of integers arises in this way; for instance, the dyadic rationals are the localization of the integers by the powers of two.
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A Survo puzzle is a kind of logic puzzle presented (in April 2006) and studied by Seppo Mustonen. Aitola, Kerttu (2006): "Survo on täällä" ("Survo is here"). Yliopisto 54(12): 44-45. The name of the puzzle is associated with Mustonen's Survo system, which is a general environment for statistical computing and related areas. Mustonen, Seppo (2007): "Survo Crossings" . CSCnews 1/2007: 30-32. In a Survo puzzle, the task is to fill an m × n table with integers 1, 2, ..., m·n so that each of these numbers appears only once and their row and column sums are equal to integers given on the bottom and the right side of the table. Often some of the integers are given readily in the table in order to guarantee uniqueness of the solution and/or for making the task easier.
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A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer. We will consider the subgroup nZ of Z consisting of all multiples of n.
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In ideal theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals.
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To make a linked list of integers, one writes `list`. A list of strings is denoted `list`. A `list` has a set of standard functions associated with it, that work for any compatible parameterizing types.
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The 7,1,3 Steane code is the first in the family of quantum Hamming codes, codes with parameters 2^r-1, 2^r-1-2r, 3 for integers r \geq 3. It is also a quantum color code.
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All solutions in integers a, b, c are given in terms of positive integer parameters m, n, k by :a=km(m+n) , \quad b=kn(m+n), \quad c=kmn where m and n are coprime.
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As an early prime number in the series of positive integers, the number seven has been associated with a great deal of symbolism in religion, mythology, superstition and philosophy. In Western culture, it is often considered lucky.
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Since any computer data can be represented as one or more machine words, one generally needs hash functions for three types of domains: machine words ("integers"); fixed-length vectors of machine words; and variable-length vectors ("strings").
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Böhm's language consisted of only assignment operations. It had no special constructs like user defined functions, control structures. Variables represented only non-negative integers. To perform a jump one had to write to a special π variable.
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As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete variables--that is, to assume only a discrete set of values, such as the integers..
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In mathematics, double brackets may also be used to denote intervals of integers or, less often, the floor function. In papyrology, following the Leiden Conventions, they are used to enclose text that has been deleted in antiquity.
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Therefore, its code points are pairs of integers 1–94. However, some encodings (UHC and Johab), in addition to providing codes for every code point, provide additional codes for characters otherwise representable only as code point sequences.
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The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... . This can be obtained by taking the coefficients of the successive powers of (1+x+x2) modulo 2 and interpreting them as integers in binary.
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See, e.g., . It is possible to find a representation of any Apollonian network as convex 3d polyhedron in which all of the coordinates are integers of polynomial size, better than what is known for other planar graphs..
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Notable 24-bit machines include the CDC 924 – a 24-bit version of the CDC 1604, CDC lower 3000 series, SDS 930 and SDS 940, the ICT 1900 series, the Elliott 4100 series, and the Datacraft minicomputers/Harris H series. The term SWORD is sometimes used to describe a 24-bit data type with the S prefix referring to sesqui. The range of unsigned integers that can be represented in 24 bits is 0 to 16,777,215 ( in hexadecimal). The range of signed integers that can be represented in 24 bits is −8,388,608 to 8,388,607.
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In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries.Dhananjay Phatak, I. Koren (1994) Hybrid Signed-Digit Number Systems: A Unified Framework for Redundant Number Representations with Bounded Carry Propagation Chains In the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead.
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For example, the integers 4, 5, 6 are (setwise) coprime (because the only positive integer dividing all of them is 1), but they are not pairwise coprime (because gcd(4, 6) = 2). The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as the Chinese remainder theorem. It is possible for an infinite set of integers to be pairwise coprime. Notable examples include the set of all prime numbers, the set of elements in Sylvester's sequence, and the set of all Fermat numbers.
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If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial: :p(t) = t − z. It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer.
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In classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integers of L can be considered as an OK[G]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e.
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Non-degenerate conic equation: :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, where at least one of the given parameters A, B, and C is non-zero, and x and y are real variables. Pell's equation: :\ x^2 - Py^2 = 1, where P is a given integer that is not a square number, and in which the variables x and y are required to be integers. The equation of Pythagorean triples: :x^2+y^2=z^2, in which the variables x, y, and z are required to be positive integers.
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Berwick was an algebraist, and worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals, and studied rings in algebraic integers. In 1927 he published Integral Bases, an ambitious account that used heavy numerical computations in place of practical proofs. He published sixteen papers, ten of them -- including a 1915 paper giving sufficient conditions for a quintic expression to be solved by radicals -- in Proceedings of the London Mathematical Society. Much of his work gained recognition only in the 1960s, when it was republished.
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The Fibonacci encodings for the positive integers are binary strings that end with "11" and contain no other instances of "11". This can be generalized to binary strings that end with N consecutive 1's and contain no other instances of N consecutive 1's. For instance, for N = 3 the positive integers are encoded as 111, 0111, 00111, 10111, 000111, 100111, 010111, 110111, 0000111, 1000111, 0100111, …. In this case, the number of encodings as a function of string length is given by the sequence of Tribonacci numbers.
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The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process.
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When and are integers, the notation ⟦a, b⟧, or or } or just , is sometimes used to indicate the interval of all integers between and included. The notation is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array. An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing , , or .
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Integers divisible by 2 are called even, and integers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non- trivial divisor (or strict divisorFoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois). A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
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These are the theories satisfying the "dimension axiom" of the Eilenberg–Steenrod axioms that the homology of a point vanishes in dimension other than 0. They are determined by an abelian coefficient group G, and denoted by H(X, G) (where G is sometimes omitted, especially if it is Z). Usually G is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p. The cohomology functors of ordinary cohomology theories are represented by Eilenberg–MacLane spaces. On simplicial complexes, these theories coincide with singular homology and cohomology.
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One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true. # Assume that is a rational number, meaning that there exists a pair of integers whose ratio is exactly . # If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
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Most Tiny BASIC interpreters (as well as Sinclair BASIC 4K) supported mathematics using integers only, lacking floating-point support. Using integers allowed numbers to be stored in a much more compact 16-bit format that could be more rapidly read and processed than the 32- or 40-bit floating- point formats found in most BASICs of the era. However, this limited its applicability as a general-purpose language. Business BASIC implementations, such as Data General Business Basic, were also integer-only, but typically at a higher precision: "double precision", i.e.
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Another optimization is to check only primes as factors in this range. For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to , which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers.
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It is the smallest number with exactly 12 divisors. It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. It is the sum of a pair of twin primes (29 + 31) and the sum of four consecutive primes (11 + 13 + 17 + 19). It is adjacent to two primes (59 and 61).
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The Set on relation instructions write one or zero to the destination register if the specified relation is true or false. These instructions source their operands from two GPRs or one GPR and a 16-bit immediate (which is sign- extended to 32 bits), and write the result to a third GPR. By default, the operands are interpreted as signed integers. The variants of these instructions that are suffixed with "unsigned" interpret the operands as unsigned integers (even those that source an operand from the sign-extended 16-bit immediate).
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There exist infinitely many non-similar triangles in which the three sides and the bisectors of each of the three angles are integers. There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers. However, for n > 3 there exist no triangles in which the three sides and the (n–1) n-sectors of each of the three angles are integers.De Bruyn,Bart, "On a Problem Regarding the n-Sectors of a Triangle", Forum Geometricorum 5, 2005: pp. 47–52.
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With angle A opposite side a and angle B opposite side b, some triangles with B=2A are generated byDeshpande,M. N., "Some new triples of integers and associated triangles", Mathematical Gazette 86, November 2002, 464–466. :a=n^2, \, :b = mn \, :c=m^2 - n^2, \, with integers m, n such that 0 < n < m < 2n. All triangles with B = 2A (whether integer or not) haveWillson, William Wynne, "A generalisation of the property of the 4, 5, 6 triangle", Mathematical Gazette 60, June 1976, 130–131. a(a+c)=b^2.
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In two's complement notation, a non-negative number is represented by its ordinary binary representation; in this case, the most significant bit is 0. Though, the range of numbers represented is not the same as with unsigned binary numbers. For example, an 8-bit unsigned number can represent the values 0 to 255 (11111111). However a two's complement 8-bit number can only represent positive integers from 0 to 127 (01111111), because the rest of the bit combinations with the most significant bit as '1' represent the negative integers −1 to −128.
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Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of th powers of positive integers is itself a th power, then is greater than or equal to : : ⇒ The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case : if , then . Although the conjecture holds for the case (which follows from Fermat's last theorem for the third powers), it was disproved for and .
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Polynomials in a single variable x can be added, multiplied and factored into irreducible polynomials, which are the analogs of the prime numbers for integers. The greatest common divisor polynomial of two polynomials and is defined as the product of their shared irreducible polynomials, which can be identified using the Euclidean algorithm. The basic procedure is similar to that for integers. At each step , a quotient polynomial and a remainder polynomial are identified to satisfy the recursive equation :r_{k-2}(x) = q_k(x)r_{k-1}(x) + r_k(x), where and .
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But notice that if we are given a particular subset we can efficiently verify whether the subset sum is zero, by summing the integers of the subset. If the sum is zero, that subset is a proof or witness for the answer is "yes". An algorithm that verifies whether a given subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time and the subset sum problem is therefore in NP. The above example can be generalized for any decision problem.
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If we want to point out specifically that the domain of integers is meant, we could write: ::∃ n ∈ ℤ; n × n = 25. Here, ∈ = is a member of... and ∈ is called the symbol for set membership; and ℤ denotes the set of integers. There are a variety of expressions that serve the same purpose in various ontologies, and they are accordingly all quantifier expressions. Quantifier variance is then one argument concerning exactly what expressions can be construed as quantifiers, and just which arguments of a quantifier, that is, which substitutions for ‘such-and-such’, are permissible.
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Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980). Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).
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The filter algorithm generalizes Peterson's algorithm to processes. Instead of a Boolean flag, it requires an integer variable per process, stored in a single writer/multiple reader (SWMR) atomic register, and additional variables in similar registers. The registers can be represented in pseudocode as arrays: level : array of N integers last_to_enter : array of N−1 integers The variables take on values up to , each representing a distinct "waiting room" before the critical section. Processes advance from one room to the next, finishing in room which is the critical section.
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Let K be a number field with ring of integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to positive or negative powers). For the ring of rational integers Z one may take S to be a finite set of prime numbers and define an S-unit to be a rational number whose numerator and denominator are divisible only by the primes in S.
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If M is a commutative monoid, it is acted on naturally by the monoid N of positive integers under multiplication, with an element n of N multiplying an element of M by n. The Frobenioid of M is the semidirect product of M and N. The underlying category of this Frobenioid is category of the monoid, with one object and a morphism for each element of the monoid. The standard Frobenioid is the special case of this construction when M is the additive monoid of non-negative integers.
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An elementary Frobenioid is a generalization of the Frobenioid of a commutative monoid, given by a sort of semidirect product of the monoid of positive integers by a family Φ of commutative monoids over a base category D. In applications the category D is sometimes the category of models of finite separable extensions of a global field, and Φ corresponds to the line bundles on these models, and the action of a positive integers n in N is given by taking the nth power of a line bundle.
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134-135); the proof in Russell 1927 PM Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plus types] must be considered as unsolved at the present time". Gödel concluded that it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (all quotes and content derived from page 135).
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1\. The torus is not homeomorphic to R2 because their fundamental groups are not isomorphic (their fundamental groups don’t have the same cardinality). More generally, a simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not. 2\. The fundamental group of the unit circle is isomorphic to the group of integers. Therefore, the one-point compactification of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the unit circle).
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Kolakoski sequences can also be created from infinite alphabets of integers, such as {1,2,1,3,1,4,1,5,...}: :1,2,2,1,1,3,1,4,4,4,1,5,5,5,5,1,1,1,1,6,6,6,6,1,7,7,7,7,7,1,1,1,1,1,8,8,8,8,8,1,1,1,1,1,9,1,10,1,11,11,11,11,11,11,... The infinite alphabet {1,2,3,4,5,...} generates the Golomb sequence: :1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,... A Kolakoski sequence can also be created from integers chosen at random from a finite alphabet, with the restriction that the same number cannot be chosen twice in a row. If the finite alphabet is {1,2,3}, one possible sequence is this: :2,2,1,1,3,1,3,3,3,2,1,1,1,2,2,2,1,1,1,3,3,2,1,3,2,2,3,3,2,2,3,1,3,1,1,1,3,3,3,1,1,3,2,2,2,3,3,1,1,3,3,3,1,1,1,3,3,1,1,2,2,2,... In effect, the sequence is based on the infinite alphabet {2,1,3,1,3,2,1,2,1,3,2,...}, which contains a random sequence of 1s, 2s and 3s from which repeats have been removed.
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IEEE 754 adds a bias to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers. If two floating-point numbers have different signs, the sign-and- magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed.
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The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers.
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The Irwin–Hall distribution is similar to the Bates distribution, but still featuring only integers as parameter. An extension to real-valued parameters is possible by adding also a random uniform variable with N − trunc(N) as width.
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For example, a program that only uses integers for arithmetic, or does no arithmetic operations at all, can exclude floating-point library routines. This smart-linking feature can lead to smaller application file sizes and reduced memory usage.
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Algebra 1 - topics range from order of operations, expressions, with variables, distributive properties, combining like terms, addition, subtraction, multiplication and division of integers, evaluate algebraic expressions, solving equations, word problem solving, inequalities, polynomials, factoring, quadratics equations and graphing.
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J. Combin. 14, 397-407, 1993. The characteristic polynomial of the Higman–Sims graph is (x − 22)(x − 2)77(x + 8)22. Therefore the Higman–Sims graph is an integral graph: its spectrum consists entirely of integers.
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The contradiction means that this assumption must be false, i.e. log2 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0\. Cases such as log10 2 can be treated similarly.
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900 (nine hundred) is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 integers. In base 10 it is a Harshad number.
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List of Fellows of the American Mathematical Society, retrieved 2013-01-19. Andrew Granville, in collaboration with Jennifer Granville, has written "Prime Suspects: The Anatomy of Integers and Permutations", a graphic novel that investigates key concepts in Mathematics.
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For example, consider to be the set of integers from to , and . Each player chooses an element of , and . Suppose player A plays and player B plays . Without loss of generality, assume player A chooses the larger number, so .
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Bjorn Poonen B. Poonen. Using elliptic curves of rank one towards the undecidability of Hilbert's tenth problem over rings of algebraic integers. In Algorithmic number theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 33-42\.
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396 = 22 × 32 × 11, sum of twin primes (197 + 199), totient sum of the first 36 integers, refactorable number, Harshad number, digit-reassembly number. 396 also refers to the displacement in cubic inches of early Chevrolet Big-Block engines.
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The extended Euclidean algorithm is the essential tool for computing multiplicative inverses in modular structures, typically the modular integers and the algebraic field extensions. A notable instance of the latter case are the finite fields of non-prime order.
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However, it is possible that two previously unequal integers will be identical modulo 9 (on average, a ninth of the time). The operation does not work on fractions, since a given fractional number does not have a unique representation.
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In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.
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For any integers and and for any prime , . The lemma is a case of the freshman's dream. Leaving the proof for later on, we proceed with the induction. Proof. Assume kp ≡ k (mod p), and consider (k+1)p.
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Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring is finitely generated as an Abelian group, which is to say, as a -module.
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Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem.
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In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element matches exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements). In the special case when is totally ordered, monotonicity of implies monotonicity of its inverse. For example, the set of integers and the set of even integers have the same order type, because the mapping n\mapsto 2n is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them.
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Any flat module is torsion- free. The converse holds over the integers, and more generally over principal ideal domains. This follows from the above characterization of flatness in terms of ideals. Yet more generally, this converse holds over Dedekind rings.
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In 2009 Marsaglia presented a version based on 64-bit integers (appropriate for 64-bit processors) which combines a multiply-with-carry generator, a Xorshift generator and a linear congruential generator. It has a period of around 2250 (around 1075).
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In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime.
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Reciprocal termial is defined as the sum of reciprocal of first positive integers. It is equal to the n-th harmonic number.Graham, R. L.; Knuth, D. E.; and Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science, 2nd ed.
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In an unsigned representation, these values are the integers between 0 and 65,535; using two's complement, possible values range from −32,768 to 32,767. Hence, a processor with 16-bit memory addresses can directly access 64 KB of byte- addressable memory.
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Springer, Berlin, 2002. has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers. Katherine Stange K. Stange.
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It is a natural extension of the circuits over sets of natural numbers when the considered set contains also negative integers, the definitions, which does not change, will not be repeated on this page. Only the differences will be mentioned.
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Pál Turán In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934.
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We say that a binary quadratic form q(x,y) represents an integer n if it is possible to find integers x and y satisfying the equation n = f(x,y). Such an equation is a representation of n by f.
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Integer addition and subtraction are computable in AC0, but multiplication is not (at least, not under the usual binary or base-10 representations of integers). Since it is a circuit class, like P/poly, AC0 also contains every unary language.
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But the programmer, especially if programming in a language supporting large integers (e.g. Lisp or Scheme), may not want wrapping arithmetic. Some architectures (e.g. MIPS), define special addition operations that branch to special locations on overflow, rather than wrapping the result.
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They vary and often deviate significantly from any geometric series in order to accommodate traditional sizes when feasible. Adjacent package sizes in these lists differ typically by factors or , in some cases even , , or some other ratio of two small integers.
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In number theory, a Williams number base b is a natural number of the form (b-1) \cdot b^n-1 for integers b ≥ 2 and n ≥ 1.Williams primes The Williams numbers base 2 are exactly the Mersenne numbers.
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The characteristic polynomial of the Nauru graph is equal to :(x-3) (x-2)^6 (x-1)^3 x^4 (x+1)^3 (x+2)^6 (x+3),\ making it an integral graph—a graph whose spectrum consists entirely of integers.
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Also, in ring theory, an element is called a "zero divisor" only if it is nonzero and for a nonzero element . Thus, there are no zero divisors among the integers (and by definition no zero divisors in an integral domain).
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Time codes are stored as 40-bit integers, which caps maximum movie length at approximately 35 years. The number of distinct streams in one file is 216, or 65536. A movie can be split into a maximum of 255 segments.
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This distribution was derived by Jacob Bernoulli. He considered the case where p = r/(r + s) where p is the probability of success and r and s are positive integers. Blaise Pascal had earlier considered the case where p = 1/2.
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One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation, polynomial long division or synthetic division.
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For example, Lebesgue measure on the real numbers is not finite, but it is σ-finite. Indeed, consider the intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line.
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The Shellsort algorithm is an sorting algorithm whose time complexity is currently an open problem. The worst case complexity has an upper bound which can be given in terms of the Frobenius number of a given sequence of positive integers.
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The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, .... This can be obtained by taking successive rows of Pascal's triangle modulo 2 and interpreting them as integers in binary, which can be graphically represented by a Sierpinski triangle.
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The only two positive integers that divide both and are and . So or . Assume , then . If does not share any elements with , then the 5 elements in besides the Identity element must be of the form where are distinct elements in .
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It includes 24 MB of flash storage, and 5.5 MB for eActivity. On this calculator, integers can be stored exactly up to approximately 22032, which is 611 digits long. After that, scientific notation is used to represent numbers up to 101000.
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255]) > Intuitionists reject the very notion of an arbitrary sequence of integers, > as denoting something finished and definite as illegitimate. Such a sequence > is considered to be a growing object only and not a finished one. (A. > Fraenkel et al.
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It is possible to prove a stronger form of Steinitz's theorem, that any polyhedral graph can be realized by a convex polyhedron for which all of the vertex coordinates are integers. For instance, Steinitz's original induction-based proof can be strengthened in this way. However, the integers that would result from this construction are doubly exponential in the number of vertices of the given polyhedral graph. Writing down numbers of this magnitude in binary notation would require an exponential number of bits.. Subsequent researchers have found lifting-based realization algorithms that use only O(n) bits per vertex... It is also possible to relax the requirement that the coordinates be integers, and assign coordinates in such a way that the x-coordinates of the vertices are distinct integers in the range [0,2n − 4] and the other two coordinates are real numbers in the range [0,1], so that each edge has length at least one while the overall polyhedron has volume O(n).. Some polyhedral graphs are known to be realizable on grids of only polynomial size; in particular this is true for the pyramids (realizations of wheel graphs), prisms (realizations of prism graphs), and stacked polyhedra (realizations of Apollonian networks)..
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Solvability means that for any three elements of a, b, x and y, the fourth exists such that the equation a x = b y is solved, hence the name of the condition. Solvability essentially is the requirement that each level P has an element in A and an element in X. Solvability reveals something about the levels of A and X — they are either dense like the real numbers or equally spaced like the integers . The Archimedean condition is as follows. Let I be a set of consecutive integers, either finite or infinite, positive or negative.
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A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used, if the writer believes that the reader can easily guess what is missing; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100.
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By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the order of the factors) factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one. For computing the factorization of an integer , one needs an algorithm for finding a divisor of or deciding that is prime. When such a divisor is found, the repeated application of this algorithm to the factors and gives eventually the complete factorization of . For finding a divisor of , if any, it suffices to test all values of such that and .
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Standard notations for relatively prime integers and are: and . In a 1989 paper, Graham, Knuth, and Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as in is prime to ). A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
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48 Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility. The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The lcm of more than two integers is also well- defined: it is the smallest positive integer that is divisible by each of them.
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As far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.
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A significant contributor to the work of unavoidable patterns, or regularities, was Frank Ramsey in 1930. His important theorem states that for integers k, m≥2, there exists a least positive integer such that despite how a complete graph is colored with two colors, there will always exist a solid color subgraph of each color. Other contributors to the study of unavoidable patterns include van der Waerden. His theorem states that if the positive integers are partitioned into k classes, then there exists a class c such that c contains an arithmetic progression of some unknown length.
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Then, for any other atom A2 with same class as A1, the vector from A1 to A2 can be written as a linear combination n u + m v, where n and m are integers. And, conversely, each pair of integers (n,m) defines a possible position for A2. Given n and m, one can reverse this theoretical operation by drawing the vector w on the graphene lattice, cutting a strip of the latter along lines perpendicular to w through its endpoints A1 and A2, and rolling the strip into a cylinder so as to bring those two points together.
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The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity. A page from Seki Takakazu's Katsuyō Sanpō (1712), tabulating binomial coefficients and Bernoulli numbers Methods to calculate the sum of the first positive integers, the sum of the squares and of the cubes of the first positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b.
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The following property holds: after h2-sorting of any h1-sorted array, the array remains h1-sorted. Every h1-sorted and h2-sorted array is also (a1h1+a2h2)-sorted, for any nonnegative integers a1 and a2. The worst-case complexity of Shellsort is therefore connected with the Frobenius problem: for given integers h1,..., hn with gcd = 1, the Frobenius number g(h1,..., hn) is the greatest integer that cannot be represented as a1h1\+ ... +anhn with nonnegative integer a1,..., an. Using known formulae for Frobenius numbers, we can determine the worst-case complexity of Shellsort for several classes of gap sequences.
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Størmer's original result can be used to show that the number of consecutive pairs of integers that are smooth with respect to a set of k primes is at most 3k − 2k. Lehmer's result produces a tighter bound for sets of small primes: (2k − 1) × max(3,(pk+1)/2). The number of consecutive pairs of integers that are smooth with respect to the first k primes are :1, 4, 10, 23, 40, 68, 108, 167, 241, 345, ... . The largest integer from all these pairs, for each k, is :2, 9, 81, 4375, 9801, 123201, 336141, 11859211, ... .
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The source of Green and Tao's arithmetic progressions is Endre Szemerédi's seminal 1975 theorem on existence of arithmetic progressions in certain sets of integers. Green and Tao showed that one can use a "transference principle" to extend the validity of Szemerédi's theorem to further sets of integers. The Green-Tao theorem then arises as a special case, although it is not trivial to show that the prime numbers satisfy the conditions of Green and Tao's extension of the Szemerédi theorem. In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions.
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In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, all range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. An important family is flavour quantum numbers – internal quantum numbers which determine the type of a particle and its interactions with other particles through the fundamental forces.
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The special case in which the integers whose reciprocals are taken must be square numbers appears in two ways in the context of right triangles. First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases.Voles, Roger, "Integer solutions of a−2+b−2=d−2," Mathematical Gazette 83, July 1999, 269-271.
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The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1. An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients). All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers.
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The model is based on treating (x,y) as equivalent to when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is :(x_p,\, x_m) \times (y_p,\, y_m) = (x_p \times y_p + x_m \times y_m,\; x_p \times y_m + x_m \times y_p) The rule that −1 × −1 = 1 can then be deduced from :(0, 1) \times (0, 1) = (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \times 0) = (1,0) Multiplication is extended in a similar way to rational numbers and then to real numbers.
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If the Erdős–Ulam problem has a positive solution, it would provide a counterexample to the Bombieri–Lang conjecture and to the abc conjecture. It would also solve Harborth's conjecture, on the existence of drawings of planar graphs in which all distances are integers. If a dense rational-distance set exists, any straight-line drawing of a planar graph could be perturbed by a small amount (without introducing crossings) to use points from this set as its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven.
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The integers with their usual topology are a discrete subgroup of the real numbers. In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H ; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.
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It is easy to check whether a value of is a solution: it suffices to compute the remainder of the Euclidean division of by each . Thus, to find the solution, it suffices to check successively the integers from to until finding the solution. Although very simple this method is very inefficient: for the simple example considered here, integers (including ) have to be checked for finding the solution, which is . This is an exponential time algorithm, as the size of the input is, up to a constant factor, the number of digits of , and the average number of operations is of the order of .
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The integers s and t of Bézout's identity can be computed efficiently using the extended Euclidean algorithm. This extension adds two recursive equations to Euclid's algorithm : : with the starting values : : Using this recursion, Bézout's integers s and t are given by s = sN and t = tN, where N+1 is the step on which the algorithm terminates with rN+1 = 0\. The validity of this approach can be shown by induction. Assume that the recursion formula is correct up to step k − 1 of the algorithm; in other words, assume that : for all j less than k.
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161 The basic principle is that each step of the algorithm reduces f inexorably; hence, if can be reduced only a finite number of times, the algorithm must stop in a finite number of steps. This principle relies on the well-ordering property of the non-negative integers, which asserts that every non-empty set of non-negative integers has a smallest member. The fundamental theorem of arithmetic applies to any Euclidean domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.
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An important condition in the theory is no small subgroups. A topological group , or a partial piece of a group like above, is said to have no small subgroups if there is a neighbourhood of containing no subgroup bigger than For example, the circle group satisfies the condition, while the -adic integers as additive group does not, because will contain the subgroups: , for all large integers . This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether can act faithfully on a closed manifold.
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In mathematics, specifically in order theory, a binary relation ≤ on a vector space X over the real or complex numbers is called Archimedean if for all x in X, whenever there exists some y in X such that nx ≤ y for all positive integers n, then necessarily x ≤ 0. An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space X is called almost Archimedean if for all x in X, whenever there exists a y in X such that -n−1y ≤ x ≤ n−1y for all positive integers n, then x = 0.
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While a 32-bit signed integer may be used to hold a 16-bit unsigned value with relative ease, a 32-bit unsigned value would require a 64-bit signed integer. Additionally, a 64-bit unsigned value cannot be stored using any integer type in Java because no type larger than 64 bits exists in the Java language. If abstracted using functions, function calls become necessary for many operations which are native to some other languages. Alternatively, it is possible to use Java's signed integers to emulate unsigned integers of the same size, but this requires detailed knowledge of complex bitwise operations.
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The polynomial : has discriminant :, and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root of it to the field of -adic numbers gives an unramified extension of . We may find the image of under the Frobenius map by locating the root nearest to , which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the -adic integers . Modulo this polynomial is :\rho^3 + 3(460+183\rho-354\rho^2-979\rho^3-575\rho^4).
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Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane (abc) where the Miller "indices" a, b and c (defined as above) are not necessarily integers. If a, b and c have rational ratios, then the same family of planes can be written in terms of integer indices (hkℓ) by scaling a, b and c appropriately: divide by the largest of the three numbers, and then multiply by the least common denominator. Thus, integer Miller indices implicitly include indices with all rational ratios.
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Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
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Euclidean domains (also known as Euclidean rings) are defined as integral domains which support the following generalization of Euclidean division: :Given an element and a non-zero element in a Euclidean domain equipped with a Euclidean function (also known as a Euclidean valuation or degree function), there exist and in such that and either or . Uniqueness of and is not required. It occurs only in exceptional cases, typically for univariate polynomials, and for integers, if the further condition is added. Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers.
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In number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, discussed in , which are additive partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been introduced by .
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Metrics are usually described in terms of variables that are a function of the input. For example, the statement that insertion sort requires O(n2) comparisons is meaningless without defining n, which in this case is the number of elements in the input list. Because many different contexts use the same letters for their variables, confusion can arise. For example, the complexity of primality tests and multiplication algorithms can be measured in two different ways: one in terms of the integers being tested or multiplied, and one in terms of the number of binary digits (bits) in those integers.
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Let us first explain why it is valid, in certain situations, to “cancel”. The exact statement is as follows. If , , and are integers, and is not divisible by a prime number , and if then we may “cancel” to obtain Our use of this cancellation law in the above proof of Fermat's little theorem was valid, because the numbers are certainly not divisible by (indeed they are smaller than ). We can prove the cancellation law easily using Euclid's lemma, which generally states that if a prime divides a product (where and are integers), then must divide or .
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The 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000. Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290.
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In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. The first odious numbers are: :1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38 ... These numbers give the positions of the nonzero values in the Thue–Morse sequence. Non-negative integers that are not odious are called evil numbers. The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums.
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A natural generalization is obtained by relaxing the condition that the coefficients of the recurrence are constants. If the coefficients are allowed to be polynomials, then one obtains holonomic sequences. A k-regular sequence satisfies linear recurrences with constant coefficients, but the recurrences take a different form. Rather than s(n) being a linear combination of s(m) for some integers m that are close to n, each term s(n) in a k-regular sequence is a linear combination of s(m) for some integers m whose base-k representations are close to that of n.
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Later, the method was extended to hashing integers by representing each byte in each of 4 possible positions in the word by a unique 32-bit random number. Thus, a table of 28x4 of such random numbers is constructed. A 32-bit hashed integer is transcribed by successively indexing the table with the value of each byte of the plain text integer and XORing the loaded values together (again, the starting value can be the identity value or a random seed). The natural extension to 64-bit integers is by use of a table of 28x8 64-bit random numbers.
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Decomposing a number n into n=x+2y, and then applying to x and y an order-preserving map from the Moser–de Bruijn sequence to the integers (by replacing the powers of four in each number by the corresponding powers of two) gives a bijection from non-negative integers to ordered pairs of non-negative integers. The inverse of this bijection gives a linear ordering on the points in the plane with non-negative integer coordinates, which may be used to define the Z-order curve.. In connection with this application, it is convenient to have a formula to generate each successive element of the Moser–de Bruijn sequence from its predecessor. This can be done as follows. If x is an element of the sequence, then the next member after x can be obtained by filling in the bits in odd positions of the binary representation of x by ones, adding one to the result, and then masking off the filled-in bits.
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In certain contexts, one may consider all sets under consideration as being subsets of some given universal set. For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset A of U, the complement of A (in U) is defined as :AC := {x ∈ U : x ∉ A}. In other words, AC ("A-complement"; sometimes simply A', "A-prime" ) is the set of all members of U which are not members of A. Thus with R, Z and O defined as in the section on subsets, if Z is the universal set, then OC is the set of even integers, while if R is the universal set, then OC is the set of all real numbers that are either even integers or not integers at all.
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He is known for his results in combinatorial number theory, and in particular for Behrend's theorem on the logarithmic density of sets of integers in which no member of the set is a multiple of any other, and for his construction of large Salem–Spencer sets of integers with no three- element arithmetic progression. Behrend sequences are sequences of integers whose multiples have density one; they are named for Behrend, who proved in 1948 that the sum of reciprocals of such a sequence must diverge. He wrote one paper in algebraic geometry, on the number of symmetric polynomials needed to construct a system of polynomials without nontrivial real solutions, several short papers on mathematical analysis, and an investigation of the properties of geometric shapes that are invariant under affine transformations. After moving to Melbourne his interests shifted to topology, first in the construction of polyhedral models of manifolds, and later in point-set topology.
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Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.
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ASN.1 is a popular binary encoding form. However, it expresses only syntax (data types), not semantics. Two different structures each a SEQUENCE of two INTEGERS have identical representations on the wire (barring special tag choices to distinguish them). To parse an ASN.
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Boole provided no proof of this rule, but the coherence of his system was proved by Theodore Hailperin, who provided an interpretation based on a fairly simple construction of rings from the integers to provide an interpretation of Boole's theory (Hailperin 1976).
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It can be shown that if G is finite, then any CW-complex modelling BG has cells of arbitrarily large dimension. On the other hand, if G = Z, the integers, then the classifying space BG is homotopy equivalent to the circle S1.
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In mathematics, the termial of a positive integer , denoted by , is the sum of all positive integers less than or equal to . For example, :5? = 5 + 4 + 3 + 2 + 1 = 15 \,. The value of is , according to the convention for an empty sum.
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The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4-space, specifically permutations of (1,2,3,4). The tessellation is formed by translated copies within the hyperplane. :240px The tessellation is the highest tessellation of parallelohedrons in 3-space.
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If the exponent here is given in NAF form, a digit value 1 implies a multiplication by the base, and a digit value −1 by its reciprocal. Other ways of encoding integers that avoid consecutive 1s include Booth encoding and Fibonacci coding.
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However in general the Exp ring can be much larger than the group ring: for example, the group ring of the integers is the ring of Laurent polynomials in 1 variable, while the exp ring is a polynomial ring in countably many generators.
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Earlier (1965) work of Michel Lazard, whose importance was not appreciated by the specialists in the field for about 20 years, had dealt with the case where K is the ring of p-adic integers and G is the pth congruence subgroup of .
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Theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate.
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Using F_{n - 2} = F_n - F_{n - 1}, one can extend the Fibonacci numbers to negative integers. So we get: :... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ... and F_{-n} = (-1)^{n + 1} F_n. See also Negafibonacci.
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An outline of the general proof and the criteria can be found in James Munkres' Topology. However, a specific case (domain is restricted to the positive integers instead of any well-ordered set) of the general recursive definition will be given below.
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Thus data can be of any type, including binary integers, floating point, or characters, without introducing a false end-of-record condition. The data set is an abstraction of a collection of records, in contrast to files as unstructured streams of bytes.
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From the properties of the binomial coefficients, it follows that the polynomials are palindromic for all positive integers , while the polynomials are palindromic when is even and antipalindromic when is odd. Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
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The mound was staked into 5-foot squares, along the cardinal directions. The northeastern stake was designated as the zero stake. The squares were designated southward by integers and westward by decimals. Stratification was not discernable and there was no evidence of intrusion.
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The mound was staked into 5-foot squares, along the cardinal directions. The northeastern stake was designated as the zero stake. The squares were designated southward by integers and westward by decimals. Stratification was not discernible and there was no evidence of intrusion.
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The most common use of symbols by programmers is for performing language reflection (particularly for callbacks), and most common indirectly is their use to create object linkages. In the most trivial implementation, they are essentially named integers (e.g. the enumerated type in C).
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720 is expressible as the product of consecutive integers in two different ways: , and . There are 49 solutions to the equation φ(x) = 720, more than any integer below it, making 720 a highly totient number. 720 is a 241-gonal number.
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This problem was given in India by the mathematician Brahmagupta in 628 AD in his treatise Brahma Sputa Siddhanta: solve the Pell's equation : x^2 - 92y^2 = 1 for integers x,y>0. Brahmagupta gave the smallest solution as : (x,y) = (1151,120).
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The following is an example on the Asmuth-Bloom's Scheme. For practical purposes we choose small values for all parameters. We choose k=3 and n=4. Our pairwise coprime integers being m_0 =3, m_1 =11, m_2 =13, m_3 =17 and m_4 =19.
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Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980. The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.
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In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authorsNiven & Zuckerman, 4.2.Nagell, I.9.Bateman & Diamond, 2.1. any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.
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Finding the quater-imaginary representation of an arbitrary real integer number can be done manually by solving a system of simultaneous equations, as shown below, but there are faster methods for both real and imaginary integers, as shown in the negative base article.
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The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)), the problem asks whether there is a labeled simple directed graph such that each vertex v_i has indegree a_i and outdegree b_i.
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Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable.J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and Other Results," Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.
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In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.. See in particular p. 99.
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Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. Furthermore, when the order is large, Fast Fourier Transformation can be used to solve for the coefficients of the interpolated polynomial.
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When the list S cannot be reduced to a list S' of nonnegative integers in any step of this approach, the theorem proves that the list S from the beginning is not graphic. The time complexity of the algorithm is O(n^2).
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Furthermore, until recently, RISC OS was only implemented on platforms whose CPUs did not afford floating point arithmetic in hardware. For that reason previous versions differed from standard Lua in only providing 32-bit integers. The current version, however, is effectively standard Lua.
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PHP stores whole numbers in a platform-dependent range. This range is typically that of 32-bit signed integers. Integer variables can be assigned using decimal (positive and negative), octal and hexadecimal notations. Real numbers are also stored in a platform-specific range.
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RNS have applications in the field of digital computer arithmetic. By decomposing in this a large integer into a set of smaller integers, a large calculation can be performed as a series of smaller calculations that can be performed independently and in parallel.
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It exists precisely when a is coprime to n, because in that case and by Bézout's lemma there are integers x and y satisfying . Notice that the equation implies that x is coprime to n, so the multiplicative inverse belongs to the group.
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The strings in this ordering over a fixed finite alphabet can be placed into one- to-one order-preserving correspondence with the non-negative integers, giving the bijective numeration system for representing numbers.. The shortlex ordering is also important in the theory of automatic groups..
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In more algebraic terms, the period lattice is a real multiple of the Gaussian integers. The constants , , and are given by :e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12. The case , may be handled by a scaling transformation. However, this may involve complex numbers.
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In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (); the sum of the next two is the next cube (); the sum of the next three is the next cube (); and so forth.
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The nth pentagonal number is the sum of n integers starting from n (i.e. from n to 2n-1). The following relationships also hold: :p_n = p_{n-1} + 3n - 2 = 2p_{n-1} - p_{n-2} + 3 Pentagonal numbers are closely related to triangular numbers.
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GraphCrunch currently supports five different types of random graph models: # Erdös-Rényi random graphs; # random graphs with the same degree distribution as the data; # Barabási-Albert preferential-attachment scale-free networks; # n-dimensional geometric random graphs (for all positive integers n); and # stickiness model networks.
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Over a field, unimodular has the same meaning as non-singular. Unimodular here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses non-singular to mean matrices that are invertible over the field.
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In mathematics, Euclid numbers are integers of the form , where pn# is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.
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After execution is completed, the instructions are held in buffers before being committed and made visible to software. Execution finishes in stage five for integer instructions and stage eight for floating- point. Committing occurs during stage six for integers, stage nine for floating-point.
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Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
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Let be the additive group of the integers, and the subgroup . Then the cosets of in are the three sets , , and , where }. These three sets partition the set ℤ, so there are no other right cosets of . Due to the commutivity of addition and .
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Later versions of the OS moved the flags to a nearby location, and Apple began shipping computers which had "32-bit clean" ROMs beginning with the release of the 1989 Mac IIci. The 68000 family stores multi-byte integers in memory in big-endian order.
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Then there are only finitely many points of E(K) whose x-coordinate is in the ring of integers OK. The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.
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Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are Fibonacci numbers and perfect numbers. For more examples, see Integer sequence.
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Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
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Combined integer and fractional values (i.e., rational numbers) can be represented by setting a radix point somewhere between two fingers (for instance, between the left and right pinkies). All digits to the left of the radix point are integers; those to the right are fractional.
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All four primitive Pythagorean quadruples with only single-digit values A Pythagorean quadruple is a tuple of integers , , and , such that . They are solutions of a Diophantine equation and often only positive integer values are considered.R. Spira, The diophantine equation , Amer. Math. Monthly Vol.
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Emma Markovna Lehmer (née Trotskaia) (November 6, 1906 - May 7, 2007) was a mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather than the more abstract aspects of the theory.
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Animation demonstrating the simplest Pythagorean triple, 32 + 42 = 52. A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer .
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The number e was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that e is irrational; that is, that it cannot be expressed as the quotient of two integers.
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In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regular sequences generalizes the class of k-automatic sequences to alphabets of infinite size.
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In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
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D. E. Knuth, "A note on solid partitions", Math. Comp., 24 (1970), 955–961. Mustonen and Rajesh extended the enumeration for all integers n\leq 50.Ville Mustonen and R. Rajesh, "Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer", J. Phys.
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This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers, and that the Euclidean algorithm allows the computation of this primitive polynomial. A consequence is that factoring polynomials over the rationals is equivalent to factoring primitive polynomials over the integers. As polynomials with coefficients in a field are more common than polynomials with integer coefficients, it may seem that this equivalence may be used for factoring polynomials with integer coefficients. In fact, the truth is exactly the opposite: every known efficient algorithm for factoring polynomials with rational coefficient uses this equivalence for reducing the problem modulo some prime number (see Factorization of polynomials).
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For factoring a multivariate polynomial over a field or over the integers, one may consider it as a univariate polynomial with coefficients in a polynomial ring with one less indeterminate. Then the factorization is reduced to factorizing separately the primitive part and the content. As the content has one less indeterminate, it may be factorized by applying the method recursively. For factorizing the primitive part, the standard method consists of substituting integers to the indeterminates of the coefficients in a way that does not change the degree in the remaining variable, factorizing the resulting univariate polynomial, and lifting the result to a factorization of the primitive part.
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Randomizing functions are used to turn algorithms that have good expected performance for random inputs, into algorithms that have the same performance for any input. For example, consider a sorting algorithm like quicksort, which has small expected running time when the input items are presented in random order, but is very slow when they are presented in certain unfavorable orders. A randomizing function from the integers 1 to n to the integers 1 to n can be used to rearrange the n input items in "random" order, before calling that algorithm. This modified (randomized) algorithm will have small expected running time, whatever the input order.
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For the large primes used in cryptography, Provable primes can be generated using variants of Pocklington primality test or Probable primes using standard probabilistic primality tests such as the Baillie–PSW primality test or the Miller–Rabin primality test. Both the provable and probable primality tests use modular exponentiation, a comparatively expensive computation. To reduce the computational cost, the integers are first checked for any small prime divisors using either sieves similar to the Sieve of Eratosthenes or Trial division. Integers with special forms, such as Mersenne prime or Fermat primes, can be efficiently tested for primality if the prime factorization of p − 1 or p + 1 is known.
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Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. :ABC conjecture III. For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε. Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc.
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Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
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The hypothesis aims to define the possible scope of a conjecture of the nature that several sequences of the type : f(n), g(n), \ldots, with values at integers n of irreducible integer-valued polynomials : f(x), g(x), \ldots, should be able to take on prime number values simultaneously, for arbitrarily large integers n. Putting it another way, there should be infinitely many such n for which each of the sequence values are prime numbers. Some constraints are needed on the polynomials. Schinzel's hypothesis builds on the earlier Bunyakovsky conjecture, for a single polynomial, and on the Hardy–Littlewood conjectures and Dickson's conjecture for multiple linear polynomials.
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Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. The bottom row of 47058 squares with side length 1/47058 is too small to see in the figure and is not shown. In number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1.
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The following presentation incorporates a slight simplification of his argument, due to Gauss, which appears in article 182 of the Disquisitiones Arithmeticae. An (integral binary) quadratic form is an expression of the form ax^2 + bxy + cy^2 with a,b,c integers. A number n is said to be represented by the form if there exist integers x,y such that n = ax^2 + bxy + cy^2. Fermat's theorem on sums of two squares is then equivalent to the statement that a prime p is represented by the form x^2 + y^2 (i.e., a=c=1, b=0) exactly when p is congruent to 1 modulo 4.
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The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if F is a field, then is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring R, the group is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group. The group K1(R) is closely related to the group of units , and if R is a field, it is exactly the group of units.
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The REDC algorithm requires products modulo R, and typically so that REDC can be used to compute products. However, when R is a power of B, there is a variant of REDC which requires products only of machine word sized integers. Suppose that positive multi-precision integers are stored little endian, that is, x is stored as an array such that for all i and . The algorithm begins with a multiprecision integer T and reduces it one word at a time. First an appropriate multiple of N is added to make T divisible by B. Then a multiple of N is added to make T divisible by B2, and so on.
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Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology. The conceptualization of rings began in the 1870s and was completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether.
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Part of Euler's motivation for studying series related to was the functional equation of the eta function, which leads directly to the functional equation of the Riemann zeta function. Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today. The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges.Euler et al.
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In Haskell, the code example filter even [1..10] evaluates to the list 2, 4, …, 10 by applying the predicate `even` to every element of the list of integers 1, 2, …, 10 in that order and creating a new list of those elements for which the predicate returns the boolean value true, thereby giving a list containing only the even members of that list. Conversely, the code example filter (not . even) [1..10] evaluates to the list 1, 3, …, 9 by collecting those elements of the list of integers 1, 2, …, 10 for which the predicate `even` returns the boolean value false (with `.` being the function composition operator).
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The sum of all -choose binomial coefficients is equal to . Consider the set of all -digit binary integers. Its cardinality is . It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with 1s (consisting of the number written as 1s). Each of these is in turn equal to the binomial coefficient indexed by and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).
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Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides and their hypotenuse are coprime integers. As a and b are not both even (otherwise and would not be coprime), one may exchange them to have even, and the parameterization is then :a = 2mn, \ \ b = m^2 - n^2, \ \ c = m^2 + n^2, where the parameters and are positive coprime integers that are not both odd. By multiplying and by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.
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Some properties of the GCD are in fact easier to see with this description, for instance the fact that any common divisor of a and b also divides the GCD (it divides both terms of ua + vb). The equivalence of this GCD definition with the other definitions is described below. The GCD of three or more numbers equals the product of the prime factors common to all the numbers, but it can also be calculated by repeatedly taking the GCDs of pairs of numbers. For example, : Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers.
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In the subtraction-based version which was Euclid's original version, the remainder calculation (`b := a mod b`) is replaced by repeated subtraction. Contrary to the division-based version, which works with arbitrary integers as input, the subtraction-based version supposes that the input consists of positive integers and stops when a = b: function gcd(a, b) while a ≠ b if a > b a := a − b else b := b − a return a The variables a and b alternate holding the previous remainders rk−1 and rk−2. Assume that a is larger than b at the beginning of an iteration; then a equals rk−2, since rk−2 > rk−1.
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Unique factorization was also a key element in an attempted proof of Fermat's Last Theorem published in 1847 by Gabriel Lamé, the same mathematician who analyzed the efficiency of Euclid's algorithm, based on a suggestion of Joseph Liouville. Lamé's approach required the unique factorization of numbers of the form , where and are integers, and is an th root of 1, that is, . Although this approach succeeds for some values of (such as , the Eisenstein integers), in general such numbers do factor uniquely. This failure of unique factorization in some cyclotomic fields led Ernst Kummer to the concept of ideal numbers and, later, Richard Dedekind to ideals.
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Diophantine definitions can be provided by simultaneous systems of equations as well as by individual equations because the system :p_1=0,\ldots,p_k=0 is equivalent to the single equation :p_1^2+\cdots+p_k^2=0. Sets of natural numbers, of pairs of natural numbers (or even of n-tuples of natural numbers) that have Diophantine definitions are called Diophantine sets. In these terms, Hilbert's tenth problem asks whether there is an algorithm to determine if a given Diophantine set is non-empty. The work on the problem has been in terms of solutions in natural numbers (understood as the non-negative integers) rather than arbitrary integers.
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Pseudorandom graphs factor prominently in the proof of the Green–Tao theorem. The theorem is proven by transferring Szemerédi's theorem, the statement that a set of positive integers with positive natural density contains arbitrarily long arithmetic progressions, to the sparse setting (as the primes have natural density 0 in the integers). The transference to sparse sets requires that the sets behave pseudorandomly, in the sense that corresponding graphs and hypergraphs have the correct subgraph densities for some fixed set of small (hyper)subgraphs. It is then shown that a suitable superset of the prime numbers, called pseudoprimes, in which the primes are dense obeys these pseudorandomness conditions, completing the proof.
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The hexagonal tortoise problem () was invented by Korean aristocrat and mathematician Choi Seok-jeong, who lived from 1646 to 1715. It is a mathematical problem that involves a hexagonal lattice, like the hexagonal pattern on some tortoises' shells, to the (N) vertices of which must be assigned integers (from 1 to N) in such a way that the sum of all integers at the vertices of each hexagon is the same. The problem has apparent similarities to a magic square although it is a vertex-magic format rather than an edge-magic form or the more typical rows-of-cells form. His book, Gusuryak, contains many interesting mathematical discoveries.
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Thus the inclusion of "partial function" extends the notion of function to "less-perfect" functions. Total- and partial-functions may either be calculated by hand or computed by machine. : Examples: :: "Functions": include "common subtraction m − n" and "addition m + n" :: "Partial function": "Common subtraction" m − n is undefined when only natural numbers (positive integers and zero) are allowed as input – e.g. 6 − 7 is undefined :: Total function: "Addition" m + n is defined for all positive integers and zero. We now observe Kleene's definition of "computable" in a formal sense: : Definition: "A partial function φ is computable, if there is a machine M which computes it" (Kleene (1952) p.
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Jacobson, I.5. p. 22 If y is the inverse of x, one can define negative powers of x by setting and (n times) for . And the rule of exponents is still verified for all integers . This is why the inverse of x is usually written .
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Advanced Tile Sets take the game of Equate to a higher mathematical level. This particular sets includes 197 tiles with positive and negative integers imprinted on them, integer exponents, fractions, the four basic operations, and equal symbols. The additional tiles are sold separately, not with the board.
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In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
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The notation can be extended into the negative exponents of the base b. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent. Numbers that are not integers use places beyond the radix point.
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This is considered the qualification round, and shots are scored as integers. In January 2018, these rules were changed so that men and women now both shoot a 3x40. The maximum qualification score is 1200. This will first be seen in the Olympic Games in 2020.
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In the design and analysis of data structures, a bucket queue (also called a bucket priority queue. See also p. 157 for the history and naming of this structure. or bounded-height priority queue) is a priority queue for prioritizing elements whose priorities are small integers.
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The homography group of the ring of integers Z is modular group . Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. The conformal group of spacetime can be represented with homographies where A is the composition algebra of biquaternions.
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Consequently, any non-Archimedean ordered field is both incomplete and disconnected. 4\. For any in the set of integers greater than has a least element. (If were a negative infinite quantity every integer would be greater than it.) 5\. Every nonempty open interval of contains a rational.
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A Bernoulli process is a discrete time process, and so the number of trials, failures, and successes are integers. Consider the following example. Suppose we repeatedly throw a die, and consider a 1 to be a "failure". The probability of success on each trial is 5/6.
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Since 2003, he has been an associate professor at the University of Paris Chevalaret. The Quillen-Lichtenbaum conjecture (from about 1971) about the relationship of the values of the Dedekind zeta function of number fields at specific locations (negative integers) is named after him and Daniel Quillen.
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Negative-base systems include negabinary, negaternary and negadecimal, with bases −2, −3, and −10 respectively; in base −b the number of different numerals used is b. Due to the properties of negative numbers raised to powers, all integers, positive and negative, can be represented without a sign.
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A Diophantine problem Diophantine analysis is the study of equations with rational coefficients requiring integer solutions. In Diophantine problems, there are fewer equatons than unknowns. The "extra" information required to solve the equations is the condition that the solutions be integers. Any solution must satisfy all equations.
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In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Viggo Brun in 1915.
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Atle Selberg In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s.
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In particular, this is the first linear-time algorithm for strings drawn from an alphabet of integers in a polynomial range. Farach's algorithm has become the basis for new algorithms for constructing both suffix trees and suffix arrays, for example, in external memory, compressed, succinct, etc.
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Fibonacci¨⍳14 ⍝ This APL statement says: Generate the Fibonacci sequence over each(¨) integer number(iota or ⍳) for the integers 1..14. 0 1 1 2 3 5 8 13 21 34 55 89 144 233 ⍝ Generated sequence, i.e., the Fibonacci sequence of numbers generated by APL's interpreter.
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The operator is flexible and may be defined arbitrarily for any given type. For example, a value of type is a range of integers, such as . is false, since the types are different (Range vs. Integer); however is true, since on values means "inclusion in the range".
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Since e and f are coprime, so are the three factors 2e, e+f, and e−f; therefore, they are each the cube of smaller integers, k, l, and m. : −2e = k3 : e + f = l3 : e − f = m3 which yields a smaller solution k3 \+ l3 \+ m3= 0.
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40 (1934). SchmidSchmid, H. L., Zyklische algebraische Funktionenkörper vom Grad pn über endlichen Konstantenkörper der Charakteristik p, Crelle 175 (1936). generalized further to non-commutative cyclic algebras of degree pn. In the process of doing so, certain polynomials related to addition of p-adic integers appeared.
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William Edward Hodgson Berwick (11 March 1888 in Dudley Hill, Bradford - 13 May 1944 in Bangor, Gwynedd) was a British mathematician, specializing in algebra, who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals.
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In terms of the Laplacian, the positive solutions to the inhomogeneous equation: :\Delta \phi = \phi - 2. The resulting numbering is unique (scale is specified by the "2"), and consists of integers; for E8 they range from 58 to 270, and have been observed as early as 1968.
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However, not every (3,6)-sparse graph is planar. Similarly, outerplanar graphs are (2,3)-sparse and planar bipartite graphs are (2,4)-sparse. Streinu and Theran show that testing (k,l)-sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2k.
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Mathematically, the problem can be formulated as follows: : Given an integer m and a set V of positive integers, find the smallest integer z that cannot be written as the sum v1 \+ v2 \+ ··· + vk of some number k ≤ m of (not necessarily distinct) elements of V.
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Order: 213 ⋅ 37 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 448345497600 Schur multiplier: Order 6. Outer automorphism group: Order 2. Other names: Sz Remarks: The 6 fold cover acts on a 12-dimensional lattice over the Eisenstein integers. It is not related to the Suzuki groups of Lie type.
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The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles. In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain.
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The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
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Because balanced ternary provides a uniform self- contained representation for integers, the distinction between signed and unsigned numerals no longer needs to be made; thereby eliminating the need to duplicate operator sets into signed and unsigned varieties, as most CPU architectures and many programming languages currently do.
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Both Java and C# support signed integers with bit widths of 8, 16, 32 and 64 bits. They use the same name/aliases for the types, except for the 8-bit integer that is called a `byte` in Java and a `sbyte` (signed byte) in C#.
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A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers. Note that the ARMA model is a univariate model. Extensions for the multivariate case are the vector autoregression (VAR) and Vector Autoregression Moving-Average (VARMA).
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38 The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.
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"Empty slots" have an offset of zero. Hashes are unsigned 32 bit integers, and start with a value of 5381. For each byte of the key, the current hash is multiplied by 33, then XOR'ed with the current byte of the key. Overflow bits are discarded.
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For example, the multiplication of two integers can be expressed as the set of triples (a, b, c) such that the relation a × b = c holds. Deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers.
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Analyst 2 believes that X follows a generalized Dirichlet distribution: X\sim GD(\alpha_1,\ldots,\alpha_k;\beta_1,\ldots,\beta_k). All parameters are again assumed to be positive integers. The analyst makes k+1 wooden boxes. The boxes have two areas: one for balls and one for marbles.
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Meg could multiply two integers in about 60 microseconds. The floating-point unit used three words for a 30-bit mantissa, and another as a 10-bit exponent. It could add two floating-point numbers in about 180 microseconds, and multiply them in about 360 μs.
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In 1776 and 1777, Felkel published a table of giving complete decompositions of all integers not divisible by 2, 3, and 5, from 1 to 408,000. Felkel had planned to extend his table to 10 million. A reconstruction of his table is found on the LOCOMAT site.
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Only supported integers. Came as standard on the Apple I and original Apple II ; Internet Basic : Written for use with the Comet system. Both were created by Signature Systems. ; IS-BASIC : The interpreter of the Enterprise 64 and 128 home computers, written by Intelligent Software Ltd.
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A second conjecture on Kummer sums was made by J. W. S. Cassels, again building on previous ideas of Tomio Kubota. This was a product formula in terms of elliptic functions with complex multiplication by the Eisenstein integers. The conjecture was proved in 1978 by Charles Matthews.
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The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol. In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers.
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A cycle basis of the graph is a set of simple cycles that forms a basis of the cycle space.. Using ideas from algebraic topology, the binary cycle space generalizes to vector spaces or modules over other rings such as the integers, rational or real numbers, etc..
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This proof, due to Euler, uses induction to prove the theorem for all integers . The base step, that , is trivial. Next, we must show that if the theorem is true for , then it is also true for . For this inductive step, we need the following lemma. Lemma.
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The first 89 natural numbers in Zeckendorf form. Each rectangle's height and width is a Fibonacci number. The vertical bands have width 10. In mathematics, Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
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Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically. There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations. The work of Atkin. et al.
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If the edge weights are integers represented in binary, then deterministic algorithms are known that solve the problem in O(m + n) integer operations.. Whether the problem can be solved deterministically for a general graph in linear time by a comparison-based algorithm remains an open question.
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We need the first condition because if the leading coefficient is negative then f(x) < 0 for all large x, and thus f(n) is not a (positive) prime number for large positive integers n. (This merely satisfies the sign convention that primes are positive.) We need the second condition because if f(x) = g(x)h(x) where the polynomials g(x) and h(x) have integer coefficients, then we have f(n) = g(n)h(n) for all integers n; but g(x) and h(x) take the values 0 and \pm 1 only finitely many times, so f(n) is composite for all large n. The third condition, that the numbers f(n) have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample. Consider f(x) = x^2 + x + 2, which has positive leading coefficient and is irreducible, and the coefficients are relatively prime; however f(n) is even for all integers n, and so is prime only finitely many times (namely when f(n)=2, in fact only at n =0,-1).
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The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The statement is about the finiteness of the set of solutions because there are 10 known solutions.
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1\. Euler's theorem can be proven using concepts from the theory of groups:Ireland & Rosen, corr. 1 to prop 3.3.2 The residue classes modulo that are coprime to form a group under multiplication (see the article Multiplicative group of integers modulo n for details). The order of that group is .
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The set of evil numbers (numbers n with t_n=0) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim- values.
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The number 2,147,483,647 (or hexadecimal 7FFFFFFF16) is the maximum positive value for a 32-bit signed binary integer in computing. It is therefore the maximum value for variables declared as integers (e.g., as `int`) in many programming languages, and the maximum possible score, money, etc. for many video games.
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For example, an interpretation I(P) of a binary predicate symbol P may be the set of pairs of integers such that the first one is less than the second. According to this interpretation, the predicate P would be true if its first argument is less than the second.
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Sixteen is an even number and a square number. Sixteen is the fourth power of two. Sixteen is the only integer that equals mn and nm, for some unequal integers m and n (m = 4, n = 2, or vice versa). It has this property because 22 = 2 × 2\.
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The general VLQ encoding is simple, but in basic form is only defined for unsigned integers (nonnegative, positive or zero), and is somewhat redundant, since prepending 0x80 octets corresponds to zero padding. There are various signed number representations to handle negative numbers, and techniques to remove the redundancy.
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Placing each vertex of the graph at the center of the corresponding circle leads to a straight line representation. Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers.; ; ; . The truth of Harborth's conjecture remains unknown .
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It states that when the condition that be logarithmically convex (or "super-convex") is added, it uniquely determines for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of .
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This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. is the exponent of the multiplicative group of integers modulo while is the order of that group. We can thus view Carmichael's theorem as a sharpening of Euler's theorem.
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Number types include integers, ratios, floating-point numbers, and complex numbers. Common Lisp uses bignums to represent numerical values of arbitrary size and precision. The ratio type represents fractions exactly, a facility not available in many languages. Common Lisp automatically coerces numeric values among these types as appropriate.
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Fermat's Last Theorem considers solutions to the Fermat equation: with positive integers , , and and an integer greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent to be a negative integer or rational, or to consider three different exponents.
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In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers.
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On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):Abramowitz and Stegun, p. 358, 9.1.5. :J_{-n}(x) = (-1)^n J_n(x). This means that the two solutions are no longer linearly independent.
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In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan WardMorgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74.
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In other words, it is a polynomial of degree seven. If , then f is a sextic function (), quintic function (), etc. The equation may be obtained from the function by setting . The coefficients may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.
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Half a circle has 180 degrees. Summing Euler's totient function φ(x) over the first + 24 integers gives 180. 180 is a Harshad number in base 10, and in binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).
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Horadam's research concerned generalised integers, formed from a sequence of real numbers greater than one (called generalised prime numbers) as the products of finite multisets of generalised primes. She was also the author of a textbook published by the University of New England, Principles of mathematics for economists (1982).
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In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by who found their asymptotic behavior, and named by , who evaluated them explicitly for the unitary group.
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The sum of the subscripts in a monomials is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers.
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The number theoretic Hilbert transform is an extension () of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences().
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Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather than a total function.
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The Reeds–Sloane algorithm, named after James Reeds and Neil Sloane, is an extension of the Berlekamp–Massey algorithm, an algorithm for finding the shortest linear feedback shift register (LFSR) for a given output sequence, for use on sequences that take their values from the integers mod n.
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By symmetry, , and is also a right isosceles triangle. It also follows that . Hence, there is an even smaller right isosceles triangle, with hypotenuse length and legs . These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms.
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The neural network consists of four weight layers: W1 (16-bit integers) and W2, W3 and W4 (8-bit). Incremental computation and single instruction multiple data (SIMD) techniques are used with appropriate intrinsic instructions, specifically in the 2018 computer shogi implementation VPADDW, VPSUBW, VPMADDUBSW, VPACKSSDW, VPACKSSWB and VPMAXSB.
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The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.
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Atoms are 32-bit integers representing strings. The protocol designers introduced atoms because they represent strings in a short and fixed size:David Rosenthal. Inter-Client Communication Conventions Manual. MIT X Consortium Standard, 1989 while a string may be arbitrarily long, an atom is always a 32-bit integer.
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The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... . This can be obtained by taking successive rows of Pascal's triangle modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
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Hofstadter also discusses the figure in his 1979 book Gödel, Escher, Bach. The structure became generally known as "Hofstadter's butterfly". David J. Thouless and his team discovered that the butterfly's wings are characterized by Chern integers, which provide a way to calculate the Hall conductance in Hofstadter's model.
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Thus it follows that together with the above ordering is an ordered ring. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.
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In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.Bourbaki, p. 116.Dummit and Foote, p. 228. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility.
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If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers up to (if any) will compute , provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions. The main practical difficulty in computing factorials is the size of the result. To assure that the exact result will fit for all legal values of even the smallest commonly used integral type (8-bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow.
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The CPU core is a two-way superscalar in-order RISC processor. Based on the MIPS R5900, it implements the MIPS-III instruction set architecture (ISA) and much of MIPS- IV, in addition to a custom instruction set developed by Sony which operated on 128-bit wide groups of either 32-bit, 16-bit, or 8-bit integers in single instruction multiple data (SIMD) fashion (i.e. four 32-bit integers could be added to four others using a single instruction). Instructions defined include: add, subtract, multiply, divide, min/max, shift, logical, leading- zero count, 128-bit load/store and 256-bit to 128-bit funnel shift in addition to some not described by Sony for competitive reasons.
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The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form , where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.Babylonian Pythagoras In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution.
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An even more conventional choice would be to label the vertices of P and Q by consecutive integers, but again there are two natural choices for how to align these labellings: Either Q_k is just clockwise from P_k or just counterclockwise. In most papers on the subject, some choice is made once and for all at the beginning of the paper and then the formulas are tuned to that choice. There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers. For this reason, the second iterate of the pentagram map is more naturally considered as an iteration defined on labeled polygons.
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The six 6th complex roots of unity form a cyclic group under multiplication. Every Abelian group can be seen as a module over the ring of integers Z, and in a finitely generated Abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators, :x = α1⋅x1 \+ α2⋅x2 \+ ... + αn⋅xn with integers α1, ..., αn. Subgroups of a finitely generated Abelian group are themselves finitely generated. The fundamental theorem of finitely generated abelian groups states that a finitely generated Abelian group is the direct sum of a free Abelian group of finite rank and a finite Abelian group, each of which are unique up to isomorphism.
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The problem of computing the number of 3-colorings of a given graph is a canonical example of a #P-complete problem, so the problem of computing the coefficients of the chromatic polynomial is #P-hard. Similarly, evaluating P(G, 3) for given G is #P-complete. On the other hand, for k=0,1,2 it is easy to compute P(G, k), so the corresponding problems are polynomial-time computable. For integers k>3 the problem is #P-hard, which is established similar to the case k=3. In fact, it is known that P(G, x) is #P-hard for all x (including negative integers and even all complex numbers) except for the three “easy points”.
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The ω in ω-model stands for the set of non-negative integers (or finite ordinals). An ω-model is a model for a fragment of second-order arithmetic whose first-order part is the standard model of Peano arithmetic, but whose second-order part may be non-standard. More precisely, an ω-model is given by a choice S⊆2ω of subsets of ω. The first-order variables are interpreted in the usual way as elements of ω, and +, × have their usual meanings, while second-order variables are interpreted as elements of S. There is a standard ω model where one just takes S to consist of all subsets of the integers.
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Different subsets of the same group can be generating subsets. For example, if p and q are integers with , then also generates the group of integers under addition by Bézout's identity. While it is true that every quotient of a finitely generated group is finitely generated (the images of the generators in the quotient give a finite generating set), a subgroup of a finitely generated group need not be finitely generated. For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = ), and let S be the subset consisting of all elements of G of the form ynxy−n for n a natural number.
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In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.
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Another definition of the GCD is helpful in advanced mathematics, particularly ring theory. The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua + vb where u and v are integers. The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer). In modern mathematical language, the ideal generated by a and b is the ideal generated by g alone (an ideal generated by a single element is called a principal ideal, and all ideals of the integers are principal ideals).
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In general, an existence proof does not provide an algorithm for computing the existing quotient and remainder, but the above proof does immediately provide an algorithm (see Division algorithm#Division by repeated subtraction), even though it is not a very efficient one as it requires as many steps as the size of the quotient. This is related to the fact that it uses only additions, subtractions and comparisons of integers, without involving multiplication, nor any particular representation of the integers such as decimal notation. In terms of decimal notation, long division provides a much more efficient algorithm for solving Euclidean divisions. Its generalization to binary and hexadecimal notation provides further flexibility and possibility for computer implementation.
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F. W. Aston subsequently discovered multiple stable isotopes for numerous elements using a mass spectrograph. In 1919 Aston studied neon with sufficient resolution to show that the two isotopic masses are very close to the integers 20 and 22, and that neither is equal to the known molar mass (20.2) of neon gas. This is an example of Aston's whole number rule for isotopic masses, which states that large deviations of elemental molar masses from integers are primarily due to the fact that the element is a mixture of isotopes. Aston similarly showed that the molar mass of chlorine (35.45) is a weighted average of the almost integral masses for the two isotopes 35Cl and 37Cl.
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Historically, a byte was the number of bits used to encode a character of text in the computer, which depended on computer hardware architecture; but today it almost always means eight bits – that is, an octet. A byte can represent 256 (28) distinct values, such as non-negative integers from 0 to 255, or signed integers from −128 to 127. The IEEE 1541-2002 standard specifies "B" (upper case) as the symbol for byte (IEC 80000-13 uses "o" for octet in French, but also allows "B" in English, which is what is actually being used). Bytes, or multiples thereof, are almost always used to specify the sizes of computer files and the capacity of storage units.
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In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's last theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am \+ bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).
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172 is an even number, a composite number and a deficient number. It is also a 30-gonal number. 172 is a noncototient integer, as well as the sum of Euler's totient function φ(x) over the first twenty-three integers. 172 is also a member of the Lazy Caterer's Sequence.
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It could multiply two 36-bit integers at a rate of 5000 per second.IBM 709 at Columbia University history page An optional hardware emulator executed old IBM 704 programs on the IBM 709. This was the first commercially available emulator. Registers and most 704 instructions were emulated in 709 hardware.
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Along with their mathematical properties, many integers have cultural significance or are also notable for their use in computing and measurement. As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.
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Same goes with 63-bit unboxed integers on 64-bit computers. Similar designs may be found in LISP and some of the other languages whose variables can take values of any type. In some cases, there was hardware support for this kind of design: see Tagged architecture and Lisp machine.
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Programming techniques are much like C using pointers to structures, occasional overlays, deliberate string handling and casts when appropriate. Available datatypes include 8 bit, 16 bit, 32 bit and (introduced later) 64 bit integers. Microcode level support was available for null terminated character strings. However, this is not commonly used.
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Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is mainly used for big integers that have a representation as a string of digits relative to some chosen numeral system base, say β = 1000 or β = 232.
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A bijective numeral system with base b uses b different numerals to represent all non-negative integers. However, the numerals have values 1, 2, 3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero.
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All primitive integer solutions (i.e., those with no prime factor common to all of a, b, and c) to the optic equation a^{-1} + b^{-1} = c^{-1} can be written asDickson, pp. 688–691. : a = mk + m^2, : b = mk + k^2, : c = mk for positive, coprime integers m, k.
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Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.
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Acta Arithmetica, vol. 27 (1975), pp. 199–245 The latter provided a positive solution to the famous Erdős–Turán conjecture from 1936 stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In a 1996 paper Bergelson and Leibman obtained an analogous statement for "polynomial progressions".
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The number of arguments within the parenthesis indicates the number of dimensions of the signal. The signal in this case is of n dimensions. A discrete signal, on the other hand, can be modeled as a function defined only on a set of points, such as the set of integers.
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Since the three factors on the right-hand side are coprime, they must individually equal cubes of smaller integers : −2e = k3 : e − 3f = l3 : e + 3f = m3 which yields a smaller solution k3 \+ l3 \+ m3= 0. Therefore, by the argument of infinite descent, the original solution (x, y, z) was impossible.
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8000 (eight thousand) is the natural number following 7999 and preceding 8001. 8000 is the cube of 20, as well as the sum of four consecutive integers cubed, 113 \+ 123 \+ 133 \+ 143. The fourteen tallest mountains on Earth, which exceed 8000 meters in height, are sometimes referred to as eight-thousanders.
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In practice, trophic levels are not usually simple integers because the same consumer species often feeds across more than one trophic level.Odum, W. E.; Heald, E. J. (1975) "The detritus- based food web of an estuarine mangrove community". Pages 265–286 in L. E. Cronin, ed. Estuarine research. Vol. 1.
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Cook, Dwork and Reischuk gave an Ω(log n) lower bound for finding the maximum of n integers allowing infinitely many processors of any PRAM without simultaneous writes.S. Cook, C. Dwork, and R. Reischuk. Upper and lower time bounds for parallel random access machines without simultaneous writes. SIAM J. Comput.
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It is known that the sequence contains infinitely many zeros and that it is unbounded. It is conjectured, but not proved, that the sequence contains every positive integer, and that every pair of non-negative integers apart from (1,1) and (n,n+1) appears as consecutive terms in the sequence.
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In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by . The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.
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Windows uses 32-bit unsigned integers as exit codes, although the command interpreter treats them as signed. If a process fails initialization, a Windows system error code may be returned. Exit codes are directly referenced, for example, by the command line interpreter CMD.exe in the `errorlevel` terminology inherited from DOS. .
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The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used Egyptian fractions . About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions.
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The real numbers with the usual ordering form a totally ordered vector space. For all integers n ≥ 0, the Euclidean space ℝn considered as a vector space over the reals with the lexicographic ordering forms a preordered vector space whose order is Archimedean if and only if n = 0 or 1.
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32 is the smallest number n with exactly 7 solutions to the equation φ(x) = n. It is also the sum of the totient function for the first ten integers. The fifth power of two, 32 is also a Leyland number since 24 \+ 42 = 32. 32 is the ninth happy number.
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A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers.
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Burnside's theorem in group theory states that if G is a finite group of order p'q, where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.
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Training consists of various tasks: addition, subtraction, multiplication, division, mixed tasks, gap-filling and comparison. Almost every task has three parts: natural numbers, integers, ratios (since 2010). More complex tasks are grouped into triathlons and pentathlons. The pentathlon consists of five tasks and 45 minutes are allowed to calculate them.
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Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if :, :, :, where are integers and , , and are prime numbers, then and are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case . Euler's rule creates additional amicable pairs for with no others being known.
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In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
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The Windows (PE/COFF) variant is based on the SysV/GNU variant. The first entry "/" has the same layout as the SysV/GNU symbol table. The second entry is another "/", a Microsoft ECOFF extension that stores an extended symbol cross-reference table. This one is sorted and uses little-endian integers.
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The packets of extensions are similar to the packets of the core protocol. The core protocol specifies that request, event, and error packets contain an integer indicating its type (for example, the request for creating a new window is numbered 1). A range of these integers are reserved for extensions.
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Gauss code, similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number.
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179 is an odd number. 179 is a prime number; that is, it is not divisible by any integer (except for 1 and itself). It is an Eisenstein prime, as it is indivisible even by complex Gaussian integers. It is a Chen prime, being two less than another prime, 181.
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This can be shown by setting equal to the rational expression a/b with a and b being positive integers with no common prime factor, and squaring to obtain N = a2/b2 and noting that since N is a positive integer b=1 so that N = a2, a square number.
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Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as , require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined.
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Making an exception for zero in the definition of evenness forces one to make such exceptions in the rules for even numbers. From another perspective, taking the rules obeyed by positive even numbers and requiring that they continue to hold for integers forces the usual definition and the evenness of zero.
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PHP is a loosely coupled language. Since, it does not depends on the data type. This is the one of main features of this language. It stores integers in a platform-dependent range, either as a 32, 64 or 128-bit signed integer equivalent to the C-language long type.
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"Omega Megasonic". Such magnetic gears, like spur gears, always have gear ratios as the ratios of small integers. More sophisticated magnetic gearing uses pole pieces to modulate the magnetic field; they can be designed to have gear ratios from 1.01:1 to 1000:1. "Magnets offer alternative to mechanical gears". 2013\.
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PushGP is a genetic programming system which evolves code written in the Push language. Push is a stack-based language designed for easy use in genetic programming, in which every variable type (e.g. strings, integers, etc.) has its own stack. All variables are stored on the stack associated with their type.
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This allows Dijkstra's algorithm to be performed in the same time bound on graphs with edges and vertices, and leads to a linear time algorithm for minimum spanning trees, with the assumption for both problems that the edge weights of the input graph are machine integers in the transdichotomous model.
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Euler established the application of binary logarithms to music theory, long before their applications in information theory and computer science became known. As part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy...
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The factorial function, generalized to all real numbers except negative integers. For example, , , . In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value where it is defined :\Pi(z) = z\Pi(z-1)\,. This is no longer a recurrence relation but a functional equation.
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But that means q1 has a proper factorization, so it is not a prime number. This contradiction shows that s does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.
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A group is called a torsion (or periodic) group if all its elements are torsion elements, and a ' if the only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z' of integers, and in this case the two notions of torsion coincide.
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But it is wrong over the real numbers, since the series does not converge. However, there are other contexts (e.g. working with 2-adic numbers, or with integers modulo a power of 2), where the series does converge. The formal calculation implies that the last equation must be valid in those contexts.
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25, 1967, p.5 allows rank functions to have arbitrary (rather than only nonnegative) integer values. In this variant, the integers can be graded (by the identity function) in his setting, and the compatibility of ranks with the ordering is not redundant. As a third variant, Brightwell and WestSee reference [2], p.722.
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They are used to encode the algebraic extensions of a finite field.Brawley & Schnibben (1989) pp.25-26 They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers. Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
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Let be a root of the primitive polynomial . The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are , , , , , , , and . The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
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Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: :Every even integer greater than 2 is the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.
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Wheel factorization with n=2x3x5=30. No primes will occur in the yellow areas. Wheel factorization is an improvement of the trial division method for integer factorization. The trial division method consists of dividing the number to be factorized successively by the first integers (2, 3, 4, 5, ...) until finding a divisor.
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Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by P_{I, M} the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
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In elementary number theory, the lifting-the-exponent (LTE) lemma provides several formulas for computing the p-adic valuation u_p of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent of p in such expressions. It is related to Hensel's lemma.
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XBasic has signed and unsigned 8-, 16- and 32-bit and signed 64-bit integers as well as 32- and 64-bit floating point values. The string data type is only for 8-bit characters. It is possible to generate an assembly language file. XBasic has a Windows only version called XBLite.
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In mathematics, the zero module is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name zero module. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.
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In this section, we consider polynomials over a unique factorization domain R, typically the ring of the integers, and over its field of fractions F, typically the field of the rational numbers, and we denote R[X] and F[X] the rings of polynomials in a set of variables over these rings.
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The tight binding energy spectrum of charged particles in a two dimensional infinite lattice is know to be self-similar and fractal, as demonstrated in Hofstadter's butterfly. For an integer ratio of the magnetic flux quantum and the magnetic flux through a lattice cell, one recovers the Landau levels for large integers.
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In modern notation, a ratio exists between quantities p and q, if there exist integers m and n such that mp>q and nq>p. This condition is known as the Archimedes property. Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal.
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In each variant of the problem, Conway uses a deck of playing cards. Since the numerical values of the deck are only relevant, only one suit is used. This is mathematically equivalent to a row of integers from 1 to N. A shuffled pile of cards is written as A[1], ..., A[N].
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Jerzy Browkin (5 November 1934 – 23 November 2015) was a Polish mathematician, studying mainly algebraic number theory. He was a professor at the Institute of Mathematics of the Polish Academy of Sciences. In 1994, together with Juliusz Brzeziński, he formulated the n-conjecture—a version of the abc conjecture involving n > 2 integers.
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On an arbitrary set , integer-valued functions form a ring with pointwise operations of addition and multiplication, and also an algebra over the ring of integers. Since the latter is an ordered ring, the functions form a partially ordered ring: :f \le g \quad\iff\quad \forall x: f(x) \le g(x).
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In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. Rings are often studied with special conditions set upon their ideals.
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In graph theory, interval edge coloring is a type of edge coloring in which edges are labeled by the integers in some interval, every integer in the interval is used by at least one edge, and at each vertex the labels that appear on incident edges form a consecutive set of distinct numbers.
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The mathematical space S of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, n-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
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Integer BASIC, as its name implies, uses integers as the basis for its math package. These were stored internally as a 16-bit number, little-endian (as is the 6502). This allowed a maximum value for any calculation between −32767 and 32767. Calculations that resulted in values outside that range produced an error.
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Euler made the first conjectures about biquadratic reciprocity.Euler, Tractatus, § 456 Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas.
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The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, .
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1-bit microprocessor MC14500BCP In computer architecture, 1-bit integers, memory addresses, or other data units are those that are (1/8 octet) wide. Also, 1-bit central processing unit (CPU) and arithmetic logic unit (ALU) architectures are those that are based on registers, address buses, or data buses of that size.
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Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
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60 However, there is a system that contains an infinite string of 9s including a last 9. The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1. The p-adic numbers are an alternative number system of interest in number theory.
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The ICI Programming Language: Frequently Asked Questions Primitive data types in ICI include integers, reals, strings, files, safe pointers, and regular expressions. Aggregate data types are arrays, sets, and associative tables. Sets can be heterogeneous, nested, and support the usual set operations: union, intersection, etc. The language supports subroutines and nested modules.
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The phase diagram of electrons in a two- dimensional square lattice, as a function of magnetic field, chemical potential and temperature, has infinitely many phases. Thouless and coworkers showed that each phase is characterized by an integral Hall conductance, where all integer values are allowed. These integers are known as Chern numbers.
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A simpler example would be any abelian Lie group. This is because any such group is a nilpotent Lie group. For example, one can take the group of real numbers under addition, and the discrete, cocompact subgroup consisting of the integers. The resulting 1-step nilmanifold is the familiar circle \R/\Z.
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Any multiple of ab (say rab) can be used to transform s and t into another solution s'=s+rb t'=t-ra: :(s+rb)a+(t-ra)b=c. Some polynomial Diophantine equations can be solved using the extended Euclidean algorithm, which works as well with polynomials as it does with integers.
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God Created the Integers: The Mathematical Breakthroughs that Changed History. Running Press. 2007. pp. 697–703.Ivan Maksimovich Lobachevsky (Jan Łobaczewski in Polish) came from a Polish noble family of Jastrzębiec and Łada coats of arms, and was classified as a Pole in Russian official documents; Jan Ciechanowicz. Mikołaj Łobaczewski - twórca pangeometrii.
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If \Gamma is any commutative monoid, then the notion of a \Gamma-graded Lie algebra generalizes that of an ordinary (\Z-) graded Lie algebra so that the defining relations hold with the integers \Z replaced by \Gamma. In particular, any semisimple Lie algebra is graded by the root spaces of its adjoint representation.
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A similar conversion between run-length-encoded binary numbers and continued fractions can also be used to evaluate Minkowski's question mark function; however, in the Calkin–Wilf tree the binary numbers are integers (positions in the breadth-first traversal) while in the question mark function they are real numbers between 0 and 1.
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In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation.
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The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C_0 of null sequences (s.th. \forall r, \exists N, \forall n > N, x_n \in H_r) is a normal subgroup of C. The factor group C/C_0 is called the completion of G with respect to H. One can then show that this completion is isomorphic to the inverse limit of the sequence (G/H_r). An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adic completion of the integers with respect to a prime p. In this case, G is the integers under addition, and Hr is the additive subgroup consisting of integer multiples of pr.
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This method works easily for finding the lcm of several integers. Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X, where X(m) is the mth iteration of X, that is, X at step m of the algorithm, etc. The purpose of the examination is to pick the least (perhaps, one of many) element of the sequence X(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as : xk(m+1) = xk(m), k ≠ k0 : xk0(m+1) = xk0(m) \+ xk0(1).
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The principal ideal theorem of class field theory states that every integer ring R (i.e. the ring of integers of some number field) is contained in a larger integer ring S which has the property that every ideal of R becomes a principal ideal of S. In this theorem we may take S to be the ring of integers of the Hilbert class field of R; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of R, and this is uniquely determined by R. Krull's principal ideal theorem states that if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.
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Presumably this is because the survey respondent assumes that negative integers refer to the presence of negative features, while smaller positive integers refer to the absence of positive features. Similarly, Schwarz has found that when a question about marital satisfaction precedes a question about general life satisfaction, responses for the two questions are highly correlated because the first question renders information about one's marriage highly accessible,Schwarz, N., Strack, F., & Mai, H. P. (1991) Assimilation and contrast effects in part-whole question sequences: A conversational logic analysis. Public Opinion Quarterly, 55, 3-23. but other studies have found the same correlation when the marital satisfaction question is asked after the general life satisfaction question, presumably because marital satisfaction is chronically accessible.
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The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra. The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole.
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In the case of fractions of integers, the fractions with and coprime and are often taken as uniquely determined representatives for their equivalent fractions, which are considered to be the same rational number. This way the fractions of integers make up the field of the rational numbers. More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, polynomials in one indeterminate, with coefficients from some integral domain D, are themselves an integral domain, call it P. So for a and b elements of P, the generated field of fractions is the field of rational fractions (also known as the field of rational functions).
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On the other hand, Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is , which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
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A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions Φt that for any element t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non- negative integers is a semi-cascade.
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Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions, ... may have multiple attributes, that fix certain properties of a number. If a number system bears the structure of an ordered ring, for example, the integers, it must contain a number that does not change any number when it is added to it (an additive identity element). This number is generally denoted as Because of the total order in this ring, there are numbers greater than zero, called the positive numbers. For other properties required within a ring, for each such positive number there exists a number less than which, when added to the positive number, yields the result These numbers less than are called the negative numbers.
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Given n polynomials with positive degrees and integer coefficients (n can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime p there is an integer x such that the values of all n polynomials at x are not divisible by p, then there are infinitely many positive integers x such that all values of these n polynomials at x are prime. For example, if the conjecture is true then there are infinitely many positive integers x such that x2 \+ 1, 3x - 1, and x2 \+ x + 41 are all prime. When all the polynomials have degree 1, this is the original Dickson's conjecture. This more general conjecture is the same as the Generalized Bunyakovsky conjecture.
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It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra. So, given an integral domain , it is often very useful to know that has a Euclidean function: in particular, this implies that is a PID.
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The two problems are equivalent: any general algorithm that can decide whether a given Diophantine equation has an integer solution could be modified into an algorithm that decides whether a given Diophantine equation has a natural number solution, and vice versa. This can be seen as follows: The requirement that solutions be natural numbers can be expressed with the help of Lagrange's four-square theorem: every natural number is the sum of the squares of four integers, so we simply replace every parameter with the sum of squares of four extra parameters. Similarly, since every integer can be written as the difference of two natural numbers, we can replace every parameter that ranges in integers with the difference of two parameters that range in natural numbers.
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In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X with its image f(X) contained in Y, so that .
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In digital signal processing, clipping occurs when the signal is restricted by the range of a chosen representation. For example in a system using 16-bit signed integers, 32767 is the largest positive value that can be represented, and if during processing the amplitude of the signal is doubled, sample values of 32000 should become 64000, but instead they are truncated to the maximum, 32767. Clipping is preferable to the alternative in digital systems — wrapping — which occurs if the digital hardware is allowed to "overflow", ignoring the most significant bits of the magnitude, and sometimes even the sign of the sample value, resulting in gross distortion of the signal. The incidence of clipping may be greatly reduced by using floating point numbers instead of integers.
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Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L. If 0# exists, then it is an example of a non-constructible Δ set of integers. This is in some sense the simplest possibility for a non- constructible set, since all Σ and Π sets of integers are constructible. On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered.
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Every possible bit combination is either a NaN or a number with a unique value in the affinely extended real number system with its associated order, except for the two combinations of bits for negative zero and positive zero, which sometimes require special attention (see below). The binary representation has the special property that, excluding NaNs, any two numbers can be compared as sign and magnitude integers (endianness issues apply). When comparing as 2's-complement integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result.
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These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a projective cover of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.
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The Actor programming language was invented by Charles Duff of The Whitewater Group in 1988. It was an offshoot of some object-oriented extensions to the Forth language he had been working on. Actor is a pure object-oriented language in the style of Smalltalk. Like Smalltalk, everything is an object, including small integers.
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In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by , that generalizes Tate duality. It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers in a number field behaves like a 3-dimensional mathematical object.
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Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is named for the mathematician Carl Friedrich Gauss. Gauss code represents a knot with a sequence of integers.
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Both of the two types of stabilizers are maximal subgroups of the whole automorphism group of the Hoffman–Singleton graph. The characteristic polynomial of the Hoffman–Singleton graph is equal to (x-7) (x-2)^{28} (x+3)^{21}. Therefore, the Hoffman–Singleton graph is an integral graph: its spectrum consists entirely of integers.
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The resulting algorithm is also known as Hopcroft–Karp algorithm. More generally, this bound holds for any unit network — a network in which each vertex, except for source and sink, either has a single entering edge of capacity one, or a single outgoing edge of capacity one, and all other capacities are arbitrary integers.
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Notice also that the set obtained by forming all the combinations a + b, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number field.
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A knot diagram with crossings labelled for a Dowker sequence In the mathematical field of knot theory, the Dowker-Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, who refined a notation originally due to Peter Guthrie Tait.
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In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups.
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The addition, subtraction and multiplication of even and odd integers obey simple rules. The addition or subtraction of two even numbers or of two odd numbers always produces an even number, e.g., 4 + 6 = 10 and 3 + 5 = 8\. Conversely, the addition or subtraction of an odd and even number is always odd, e.g.
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The simple modules are the modules Es/r where r and s are coprime integers with r>0. The module Es/r has a basis over W(k)[1/p] of the form v, Fv, F2v,...,Fr−1v for some element v, and . The rational number s/r is called the slope of the module.
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In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.
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Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of in , the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.
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The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to in this statement, since Z is isomorphic to Z/0Z.
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The operations + and ⋅ are called addition and multiplication, respectively. The multiplication symbol ⋅ is usually omitted; for example, xy means . Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not necessarily equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
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In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to n in which no member of the set is a multiple of any other must have a logarithmic density that goes to zero as n becomes large. The theorem is named after Felix Behrend, who published it in 1935.
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However, it is defined for all other complex numbers. This definition is consistent with the earlier definition only for those integers satisfying . In addition to extending to most complex numbers , this definition has the feature of working for all positive real values of . Furthermore, when , this definition is mathematically equivalent to the function, described above.
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Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers only occur in certain combinations of values, and their physical interpretation changes depending on whether real or complex versions of the atomic orbitals are employed.
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Java does not feature unsigned integer types. In particular, Java lacks a primitive type for an unsigned byte. Instead, Java's `byte` type is sign extended, which is a common source of bugs and confusion. Unsigned integers were left out of Java deliberately because James Gosling believed that programmers would not understand how unsigned arithmetic works.
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The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. ElGamal encryption can be defined over any cyclic group G, like multiplicative group of integers modulo n. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms.
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Every finite group is finitely generated since . The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. For example, the group of real numbers under addition, (R, +).
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All five of these numbers play important and recurring roles across mathematics, and these five constants appear in one formulation of Euler's identity. Like the constant , is also irrational (i.e. it cannot be represented as ratio of integers) and transcendental (i.e. it is not a root of any non-zero polynomial with rational coefficients).
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In a cluster, the explicit type conversions up and down change between the abstract type and the representation. There is a universal type any, and a procedure force[] to check that an object is a certain type. Objects may be mutable or immutable, the latter being base types such as integers, booleans, characters and strings.
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The number of indices needed to specify an element is called the dimension, dimensionality, or rank of the array. In standard arrays, each index is restricted to a certain range of consecutive integers (or consecutive values of some enumerated type), and the address of an element is computed by a "linear" formula on the indices.
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An element of a ring that is equal to its own square is called an idempotent. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo has idempotents, where is the number of distinct prime factors of .
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If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
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Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.
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All Pythagorean quadruples (including non-primitives, and with repetition, though , and do not appear in all possible orders) can be generated from two positive integers and as follows: If and have different parity, let be any factor of such that . Then and . Note that . A similar method existsSierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig.
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Taylor was born in Leicester in 1952 and educated at Wyggeston Grammar School. He gained a first class degree from Pembroke College, Oxford in 1973, and a Ph.D. from King's College London with a thesis entitled Galois module structure of the ring of integers of l-extensions in 1976 under the supervision of Albrecht Fröhlich.
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The widely used two's complement encoding does not allow a negative zero. In a 1+7-bit sign-and-magnitude representation for integers, negative zero is represented by the bit string . In an 8-bit one's complement representation, negative zero is represented by the bit string . In all three encodings, positive zero is represented by .
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It has a graph store to hold the Clean graph that is being rewritten. The A(rgument)-stack holds arguments that refer to nodes in the graph store. This way, a node's arguments can be rewritten, which is needed for pattern matching. The B(asic value)-stack holds basic values (integers, characters, reals, etc.).
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Scalar processors represent a class of computer processors. A scalar processor processes only one data item at a time, with typical data items being integers or floating point numbers.Advanced Microprocessors and Interfacing by Badri Ram 2000 page 11 A scalar processor is classified as a SISD processor (Single Instructions, Single Data) in Flynn's taxonomy.
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Fifty-eight is the sum of the first seven prime numbers, an 11-gonal number, and a Smith number. Given 58, the Mertens function returns 0. There is no solution to the equation x – φ(x) = 58, making 58 a noncototient. However, the sum of the totient function for the first thirteen integers is 58.
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To say that the field of rational numbers is infinitely divisible (i.e. order theoretically dense) means that between any two rational numbers there is another rational number. By contrast, the ring of integers is not infinitely divisible. Infinite divisibility does not imply gap-less-ness: the rationals do not enjoy the least upper bound property.
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Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).
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It is unknown whether this necessary condition is sufficient. Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non- zero natural density.
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For most data types, specific variation operators can be designed. Different chromosomal data types seem to work better or worse for different specific problem domains. When bit-string representations of integers are used, Gray coding is often employed. In this way, small changes in the integer can be readily affected through mutations or crossovers.
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The results of the preceding section remain valid if the ring of integers and the field of rationals are respectively replaced by any unique factorization domain and its field of fractions . This is typically used for factoring multivariate polynomials, and for proving that a polynomial ring over a unique factorization domain is also a unique factorization domain.
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Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion. Defining to be the ring of integers of a totally real algebraic number field , one then defines the Hilbert–Blumenthal modular group as . One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.
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William Burnside. In mathematics, Burnside's theorem in group theory states that if G is a finite group of order p^a q^b where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non- Abelian finite simple group has order divisible by at least three distinct primes.
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Given an integral domain , its field of fractions is built with the fractions of two elements of exactly as Q is constructed from the integers. More precisely, the elements of are the fractions where and are in , and . Two fractions and are equal if and only if . The operation on the fractions work exactly as for rational numbers.
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In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general.
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In parallel with route numbering, the junctions between routes and exits from routes are also labelled with exit numbers. On every route, exits are numbered from one end to the other with ascending consecutive integers with a mixture of alphabet-suffixed labels (1, 2, 2A, 2B, 3, 4... etc.), similar in function to UK motorway junction markers.
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Tarski sketched the (nontrivial) proof of how these axioms and primitives imply the existence of a binary operation called multiplication and having the expected properties, so that R is a complete ordered field under addition and multiplication. This proof builds crucially on the integers with addition being an abelian group and has its origins in Eudoxus' definition of magnitude.
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The integers k,s,m and the real numbers \sigma_i are uniquely determined. Note that k+s+m=d. The factor I_m \oplus 0_s corresponds to the maximal invariant subspace on which P acts as an orthogonal projection (so that P itself is orthogonal if and only if k=0) and the \sigma_i-blocks correspond to the oblique components.
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Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally likely outcomes. These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: 2:5 = (2/7):(5/7). Conversely, the odds against is the opposite ratio.
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Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.
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For example, you can have a parameter that is an array of five integers. The parameters/result structure and the set of data types are meant to mirror those used in common programming languages. Identification of clients for authorization purposes can be achieved using popular HTTP security methods. Basic access authentication can be used for identification and authentication.
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A Euclidean domain is an integral domain on which is defined a Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. In a Euclidean domain, Euclidean division allows defining a Euclidean algorithm for computing greatest common divisors. However this does not imply the existence of a factorization algorithm.
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Encoding schemes are used to convert coordinate integers into binary form to provide additional compression gains. Encoding designs, such as the Golomb code and the Huffman code, have been incorporated into genomic data compression tools. Of course, encoding schemes entail accompanying decoding algorithms. Choice of the decoding scheme potentially affects the efficiency of sequence information retrieval.
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Daniel Quillen showed that the Bass conjecture holds for all (regular, depending on the version of the conjecture) rings or schemes of dimension ≤ 1, i.e., algebraic curves over finite fields and the spectrum of the ring of integers in a number field. The (non-regular) ring A = Z[x, y]/x2 has an infinitely generated K1(A).
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If M is a transitive model, then ωM is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each real number in a transitive model is a standard real number, although not all standard reals need be included in a particular transitive model.
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A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.
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Numbers such as and are said to be associate. In the integers, the primes and are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity.
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But Gardner was mistaken about the difficulty. In fact, if (N, F) is a solution then so is (N + 15625 t, F + 1024 t) for any integer t. This means that the equation also has solutions in negative integers. Trying out a few small negative numbers it turns out N = -4 and F = -1 is a solution.
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In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers } generated by the family of all arithmetic progressions. It is a special case of the profinite topology on a group. This particular topological space was introduced by where it was used to prove the infinitude of primes.
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In general topology, a branch of mathematics, the Appert topology, named for , is a topology on the set } of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.
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Dynamic Parallelism and HyperQ, two features in GK110/GK208 GPUs, are also supported across the entire Maxwell product line. Maxwell provides native shared memory atomic operations for 32-bit integers and native shared memory 32-bit and 64-bit compare-and-swap (CAS), which can be used to implement other atomic functions. Maxwell supports DirectX 12.
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With the same input as in the preceding sections, the successive remainders, after division by their content are :-5\,X^4+X^2-3, :13\,X^2+25\,X-49, :4663\,X-6150, :1. The small size of the coefficients hides the fact that a number of integers GCD and divisions by the GCD have been computed.
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Borel's game is similar to the above example for very large S, but the players are not limited to round integers. They thus have an infinite number of available pure strategies, indeed a continuum. This concept is also implemented in a story of Sun Bin when watching a chariot race with three different races running concurrently.
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Maxwell provides native shared memory atomic operations for 32-bit integers and native shared memory 32-bit and 64-bit compare-and-swap (CAS), which can be used to implement other atomic functions. While it was once thought that Maxwell used tile-based immediate mode rasterization, Nvidia corrected this at GDC 2017 saying Maxwell instead uses Tile Caching.
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The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes . More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates) . The elements of a set may be manifested in music as simultaneous chords, successive tones (as in a melody), or both.
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242 is the smallest integer to start a run of four consecutive integers with the same number of divisors.R. K. Guy Unsolved Problems in Number Theory, section B18.D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986: 147, 176. 242 is a nontotient since there is no integer with 242 coprimes below it.
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All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract algebra. Performing subtraction on natural numbers is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children.
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It turns out that this is the only way that an orbit can contain infinitely many integers. :Theorem. Let be a rational function of degree at least two, and assume that no iterateAn elementary theorem says that if and if some iterate of is a polynomial, then already the second iterate is a polynomial. of is a polynomial. Let .
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Given a matrix and a matrix whose components are all integers, Green and Tao give conditions on when there exist infinitely many matrices such that all components of are prime numbers. The proof of Green and Tao was incomplete, as it was conditioned upon unproven conjectures. Those conjectures were proved in later work of Green, Tao, and Tamar Ziegler.
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A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if X has a Poisson distribution with expected value λ then the variance of X is also λ, and :P(X\leq x)=P(X if Y is normally distributed with expectation and variance both λ.
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It is known that an odd almost perfect number greater than 1 would have at least six prime factors. If m is an odd almost perfect number then is a Descartes number. Moreover if a and b are positive odd integers such that b+3 and such that and are both primes, then would be an odd weird number.
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Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.
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His mathematical libraries created for the GIMPS project are the fastest known for multiplication of large integers, and are used by other distributed computing projects as well, such as Seventeen or Bust. He also worked on a TTL version of Maze War while a student at MIT. Later he worked as a programmer for Data General.
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Small Eisenstein primes. If and are Eisenstein integers, we say that divides if there is some Eisenstein integer such that . A non-unit Eisenstein integer is said to be an Eisenstein prime if its only non-unit divisors are of the form , where is any of the six units. There are two types of Eisenstein primes.
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C# supports unsigned in addition to the signed integer types. The unsigned types are `byte`, `ushort`, `uint` and `ulong` for 8, 16, 32 and 64 bit widths, respectively. Unsigned arithmetic operating on the types are supported as well. For example, adding two unsigned integers (`uint`s) still yields a `uint` as a result; not a long or signed integer.
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Given S = {3,1,1,2,2,1}, a valid solution to the partition problem is the two sets S1 = {1,1,1,2} and S2 = {2,3}. Both sets sum to 5, and they partition S. Note that this solution is not unique. S1 = {3,1,1} and S2 = {2,2,1} is another solution. Not every multiset of positive integers has a partition into two subsets with equal sum.
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Faulhaber made the first publication of Henry Briggs's Logarithm in Germany. He's also credited with the first printed solution of equal temperament.Date,name,ratio,cents: from equal temperament monochord tables p55-p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951 He died in Ulm. Faulhaber's major contribution was in calculating the sums of powers of integers.
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In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles are either tangent or disjoint from each other.
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If is a multiple of , then for some integer , and is a solution. On the other hand, for every pair of integers and , the greatest common divisor of and divides . Thus, if the equation has a solution, then must be a multiple of . If and , then for every solution , we have :, showing that is another solution.
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A one-dimensional array (or single dimension array) is a type of linear array. Accessing its elements involves a single subscript which can either represent a row or column index. As an example consider the C declaration `int anArrayName[10];` which declares a one-dimensional array of ten integers. Here, the array can store ten elements of type `int` .
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The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers. Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing.
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Kramp could read and write in German and French. Kramp was appointed professor of mathematics at Strasbourg, the town of his birth, in 1809. He was elected to the geometry section of the French Academy of Sciences in 1817. As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers.
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1\. Since the ordered set: A = (−∞, 0) U (0,+∞) is not a linear continuum, it is disconnected. 2\. By applying the theorem just proved, the fact that R is connected follows. In fact any interval (or ray) in R is also connected. 3\. The set of integers is not a linear continuum and therefore cannot be connected. 4\.
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As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring ).See Euler's theorem.
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In general, the negativity or positivity of a number is referred to as its sign. Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers.
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The easiest way to do the addition is to translate the numbers to quote notation, then add, then translate back. Likewise for subtraction. To add two numbers in quote notation, just add them the same way you add two positive integers. The repetition is recognized when the repeating parts of the two operands return to their starting digits.
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When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein's criterion can also be used in some cases to determine irreducibility.
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In some languages, this operator is referred to as the conditional operator. The multiply–accumulate operation is another ternary operator. Another example of a ternary operator is between, as used in SQL. The Icon programming language has a "to-by" ternary operator: the expression `1 to 10 by 2` generates the odd integers from 1 through 9.
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However, not all powerful numbers are Achilles numbers: only those that cannot be represented as , where and are positive integers greater than 1. Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers.
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In C++03, enumerations are not type-safe. They are effectively integers, even when the enumeration types are distinct. This allows the comparison between two enum values of different enumeration types. The only safety that C++03 provides is that an integer or a value of one enum type does not convert implicitly to another enum type.
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In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by and . In Russia it is sometimes called the A-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A).
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Albert Girard was the first to make the observation, describing all positive integer numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625.Simon Stevin. l'Arithmétique de Simon Stevin de Bruges, annotated by Albert Girard, Leyde 1625, p. 622.L. E. Dickson, History of the Theory of Numbers, Vol.
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Unsigned integers are converted to signed values in certain situations, which is different behavior to many other programming languages. Integer variables can be assigned using decimal (positive and negative), octal, hexadecimal, and binary notations. Floating point numbers are also stored in a platform-specific range. They can be specified using floating point notation, or two forms of scientific notation.
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The divide operation worked similarly. It also had an integer multiplication but, because the accumulator had 32 bits while memory words had only 31 bits, only even integers could be thus represented. To further reduce costs, the traditional front panel lights showing internal registers were absent. Instead, Librascope mounted a small oscilloscope on the front panel.
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123, No. 8, October 2016, pp. 753-776" [This paper describes work of Erdős, Klarner, and Rado on semigroups of integer affine maps and on sets of integers they generate. It gives the history of problems they studied, some solutions, and new unsolved problems that arose from them."] The Klarner-Rado Sequence is named after Klarner and Richard Rado.
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Unlike the above example, the character classification routines are not written as comparison tests. In most C libraries, they are written as static table lookups instead of macros or functions. For example, an array of 256 eight-bit integers, arranged as bitfields, is created, where each bit corresponds to a particular property of the character, e.g., isdigit, isalpha.
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Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers.
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Kempner showed the sum of this series is less than 80. Baillie showed that, rounded to 20 decimals, the actual sum is . Heuristically, this series converges because most large integers contain every digit. For example, a random 100-digit integer is very likely to contain at least one '9', causing it to be excluded from the above sum.
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In computer science, an (a,b) tree is a kind of balanced search tree. An (a,b)-tree has all of its leaves at the same depth, and all internal nodes except for the root have between and children, where and are integers such that . The root has, if it is not a leaf, between 2 and children.
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The latter come as function fields of schemes of finite type over integers and their appropriate localization and completions. The theory is referred to as higher local class field theory and higher global class field theory. It uses algebraic K-theory and appropriate Milnor K-groups replace K_1 which is in use in one-dimensional class field theory.
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In graph theory, a recursive tree (i.e., unordered tree) is a non-planar labeled rooted tree. A size-n recursive tree is labeled by distinct integers 1, 2, ..., n, where the labels are strictly increasing starting at the root labeled 1. Recursive trees are non-planar, which means that the children of a particular node are not ordered. E.g.
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The interpretation of any value was determined by the operators used to process the values. (For example, `+` added two values together, treating them as integers; `!` indirected through a value, effectively treating it as a pointer.) In order for this to work, the implementation provided no type checking. Hungarian notation was developed to help programmers avoid inadvertent type errors.
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The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples consist of all red or all blue members. The Boolean Pythagorean triples problem was solved by Marijn Heule, Oliver Kullmann and Victor W. Marek in May 2016 through a computer-assisted proof.
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2D diagram of mellitate , one of the oxocarbon anions. Black circles are carbon atoms, red circles are oxygen atoms. Each blue halo represents one half of a negative charge. In chemistry, an oxocarbon anion is a negative ion consisting solely of carbon and oxygen atoms, and therefore having the general formula for some integers x, y, and n.
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Notably, while the first-generation TPUs were limited to integers, the second- generation TPUs can also calculate in floating point. This makes the second- generation TPUs useful for both training and inference of machine learning models. Google has stated these second-generation TPUs will be available on the Google Compute Engine for use in TensorFlow applications.
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Some programming languages also permit digit group separators. The internal representation of this datum is the way the value is stored in the computer's memory. Unlike mathematical integers, a typical datum in a computer has some minimal and maximum possible value. The most common representation of a positive integer is a string of bits, using the binary numeral system.
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An expression of the form Q(x,y)=ax^2+bxy+cy^2, where a, b and c are fixed integers and x and y are variable integers, is called an integer binary quadratic form. The discriminant of the form is defined as :D = b^2 -4ac. The form is said to be primitive if the coefficients a, b, c are relatively prime. Two forms :Q(x,y) = ax^2+bxy+cy^2, \quad Q^\prime(x,y)=a^\prime x^2+b^\prime xy + c^\prime y^2 are said to be equivalent if there exists a transformation :x\mapsto \alpha x + \beta y,\quad y\mapsto \gamma x + \delta y with integer coefficients satisfying \alpha\delta - \beta\gamma =1 which transforms Q(x,y) to Q^\prime(x,y).
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The concept of theories of arithmetic whose integers are the true mathematical integers is captured by ω-logic.J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. Let T be a theory in a countable language which includes a unary predicate symbol N intended to hold just of the natural numbers, as well as specified names 0, 1, 2, ..., one for each (standard) natural number (which may be separate constants, or constant terms such as 0, 1, 1+1, 1+1+1, ..., etc.). Note that T itself could be referring to more general objects, such as real numbers or sets; thus in a model of T the objects satisfying N(x) are those that T interprets as natural numbers, not all of which need be named by one of the specified names.
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The method first selects a primorial and then constructs an interval in which the distribution of integers coprime to the primorial is well understood. By looking at copies of the interval translated by multiples of the primorial an array (or matrix) of integers is formed where the rows are the translated intervals and the columns are arithmetic progressions where the difference is the primorial. By Dirichlet's theorem on arithmetic progressions the columns will contain many primes if and only if the integer in the original interval was coprime to the primorial. Good estimates for the number of small primes in these progressions due to allows the estimation of the primes in the matrix which guarantees the existence of at least one row or interval with at least a certain number of primes.
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In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, an infinite arithmetic progression is full if there exist integers a and b such that the progression consists of all positive integers equal to b modulo a. This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0).
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The B register is used for indexing into arrays; the LNB (Local Name Base) register points to the base of the current stack frame, with the SF (Stack Front) register pointing to the movable 'top' of the stack; the DR register is used for holding descriptors for addressing into the heap, and so on. There are also two 32 bit pointers to off-stack data; XNB (eXtra Name Base) and LTB (Linkage Table Base). Data formats recognized by the PLI instructions include 32-bit unsigned integers; 32-bit and 64-bit twos-complement integers; 32-bit, 64-bit and 128-bit floating point; and 32-bit, 64-bit, and 128-bit packed decimal. Conventionally (and strangely to those tutored on C and UNIX) the boolean value true is represented as zero, false as minus one.
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For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising in this fashion are typically referred to as analytic p-adic L-functions. The other major source of p-adic L-functions--first discovered by Kenkichi Iwasawa--is from the arithmetic of cyclotomic fields, or more generally, certain Galois modules over towers of cyclotomic fields or even more general towers. A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved.
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The residue class ring modulo a Gaussian integer is a field if and only if z_0 is a Gaussian prime. If is a decomposed prime or the ramified prime (that is, if its norm is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, ). It is thus isomorphic to the field of the integers modulo . If, on the other hand, is an inert prime (that is, is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with elements (the integers modulo ).
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Nodes are of seven kinds, corresponding to different constructs in the syntax of XML: elements, attributes, text nodes, comments, processing instructions, namespace nodes, and document nodes. (The document node replaces the root node of XPath 1.0, because the XPath 2.0 model allows trees to be rooted at other kinds of node, notably elements.) Nodes may be typed or untyped. A node acquires a type as a result of validation against an XML Schema. If an element or attribute is successfully validated against a particular complex type or simple type defined in a schema, the name of that type is attached as an annotation to the node, and determines the outcome of operations applied to that node: for example, when sorting, nodes that are annotated as integers will be sorted as integers.
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In algebra, Gauss's lemma,Article 42 of Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801) named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers.
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Scale factors are also used in floating-point numbers, and most commonly are powers of two. For example, the double-precision format sets aside 11 bits for the scaling factor (a binary exponent) and 53 bits for the significand, allowing various degrees of precision for representing different ranges of numbers, and expanding the range of representable numbers beyond what could be represented using 64 explicit bits (though at the cost of precision). As an example of where precision is lost, a 16-bit unsigned integer (uint16) can only hold a value as large as 65,53510. If unsigned 16-bit integers are used to represent values from 0 to 131,07010, then a scale factor of would be introduced, such that the scaled values correspond exactly to the real-world even integers.
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Because they are closed under addition, subtraction, and multiplication, but not division, the p-adic fractions are a ring but not a field. As a ring, the p-adic fractions are a subring of the rational numbers Q, and an overring of the integers Z. Algebraically, this subring is the localization of the integers Z with respect to the set of powers of p. The set of all p-adic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form \lfloor 2^i x \rfloor / 2^i. Compared to other dense subsets of the real line, such as the rational numbers, the p-adic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs.
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While a 32-bit signed integer may be used to hold a 16-bit unsigned value losslessly and a 32-bit unsigned value would require a 64-bit signed integer, a 64-bit unsigned value cannot be stored easily using any integer type because no type larger than 64 bits exists in the Java language. In all cases, the memory consumed may increase by a factor of up to two, and any logic that depends on the rules of two's complement overflow must typically be rewritten. If abstracted using functions, function calls become necessary for many operations which are native to some other languages. Alternatively, it is possible to use Java's signed integers to emulate unsigned integers of the same size, but this requires detailed knowledge of bitwise operations.
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In computer programming, a handle is an abstract reference to a resource that is used when application software references blocks of memory or objects that are managed by another system like a database or an operating system. A resource handle can be an opaque identifier, in which case it is often an integer number (often an array index in an array or "table" that is used to manage that type of resource), or it can be a pointer that allows access to further information. Common resource handles include file descriptors, network sockets, database connections, process identifiers (PIDs), and job IDs. PIDs and job IDs are explicitly visible integers; while file descriptors and sockets (which are often implemented as a form of file descriptor) are represented as integers, they are typically considered opaque.
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" He was the first to give a general solution to the linear Diophantine equation ax + by = c, where a, b, and c are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave all integer solutions; but that Brahmagupta used some of the same examples as Diophantus has led some historians to consider the possibility of a Greek influence on Brahmagupta's work, or at least a common Babylonian source. "he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [...] It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation.
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In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
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Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: :Every computably enumerable set is Diophantine. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0. Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0.
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In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules: :(a, b) + (c, d) = (a + c, b + d) :(a, b) × (c, d) = (a × c + b × d, a × d + b × c) We define an equivalence relation ~ upon these pairs with the following rule: :(a, b) ~ (c, d) if and only if a + d = b + c. This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.
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That satisfaction of Euclid's formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers m and n, m > n, the a, b, and c given by the formula are all positive integers, and from the fact that : a^2+b^2 = (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 = c^2. A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such triples can be written as (a, b, c) where and a, b, c are coprime. Thus a, b, c are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one).
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If the coefficients do not belong to Fp, the p-th root of a polynomial with zero derivative is obtained by the same substitution on x, completed by applying the inverse of the Frobenius automorphism to the coefficients. This algorithm works also over a field of characteristic zero, with the only difference that it never enters in the blocks of instructions where pth roots are computed. However, in this case, Yun's algorithm is much more efficient because it computes the greatest common divisors of polynomials of lower degrees. A consequence is that, when factoring a polynomial over the integers, the algorithm which follows is not used: one compute first the square-free factorization over the integers, and to factor the resulting polynomials, one chooses a p such that they remain square-free modulo p.
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