Phil Carmody discovered a 1,401-digit prime number—no, we're not going to post it—that (with the right know-how) was executable as the very same illegal software—hence, an illegal prime number.
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Among others, I got PRIME NUMBER, FAX NUMBER and WRONG NUMBER.
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"11 is my favorite prime number," Porowski wrote alongside the photo.
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But not all integers plugged into this expression generate a prime number.
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The classic case is a prime number produced by Carmody in 2001.
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The typical notion of a prime number doesn't make sense for finite fields.
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A prime number is one that can be divided by only 1 and itself.
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And then, because we're all computer geeks, we loved the whole "prime number" thing.
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Simply, this is a prime number that represents information that is illegal to possess.
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The odds of one of my computers making a prime number discovery are astronomical.
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It's the 49th Mersenne prime number discovered since the search began in about 500 BCE.
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Before that it was 2.5 and then 9.4, which is not a prime number, but. Okay.
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Adding a prime number gear for the fulcrum makes the drawing into a non-repeating line.
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GIMPS discovered the prime number by multiplying 20153 by itself 74,207,281 times and then subtracting 1.
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The prime number sequences Tarkhanov works with in beweistheorie I are: 562613, 562621, 562631, 565626313, 562651, and 562663.
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The largest known prime number, newly discovered, is almost five million digits longer than the previous record-holder.
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It looks like someone has imprinted a series of spikes into the data, following the prime number series.
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The background software running on the computer unearthed a rare kind of prime number called a Mersenne prime.
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As we explained then: A prime number is not divisible by any positive integer except 1 and itself.
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A FedEx employee from Tennessee has made mathematical history with the discovery of the largest known prime number.
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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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In an "ode" to its Prime members, the number on the tail of the aircraft is a prime number.
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If you're wondering: If a prime number is discovered and no one is there to notice, is it really discovered?
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Riemann's formula works for the first 10 trillion solutions, said the institute, but not for every single potential prime number.
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A few of the things you'll be able to ask Wolfram Alpha via Alexa: Alexa, what is the billionth prime number?
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A prime number discovery in December was made in the unlikeliest of places: on a church computer in a Memphis suburb.
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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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Why does the simple act of combining the greatest weekday with the worst prime number strike such fear into our hearts?
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But music that has never been heard on Earth before… That would rank up there with the prime number sequence from Contact.
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Similarly, at 49A, we have PRIME NUMBER, which contains the SECRET MENU that I never seem to be successful at ordering from.
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As exciting as this discovery is, the Holy Grail of primes is yet to be found: a prime number containing 100 million digits.
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It then checked that this number was not divisible by any positive integer except 1 and itself — the definition of a prime number.
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The twin primes conjecture for finite fields concerns polynomials with just one factor (just as a prime number has a single factor—itself).
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The largest prime number in the world has been discovered in Missouri by the Great Internet Mersenne Prime Search project, better known as GIMPS.
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Luckily, not even enough time passed between the mechanical discovery and the real one to say the 22 million prime number in its entirety.
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Put in n = 2338, and the result is 2618⁴ − 22 = 21, which is not a prime number, because 26 is divisible by 73 and 27.
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The new record-holding prime number, dubbed "M77232917," was discovered by Jonathan Pace, a 51-year-old electrical engineer living in Germantown, Tennessee, on December 26, 2017.
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The PC that Pace used to find the prime number required six straight days of computation on a quad-core Intel i5-6600 CPU to verify it.
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ShutterstockUsing a computer powered by an off-the-shelf Intel Core i5-6600 processor, a FedEx employee from Tennessee has discovered the largest prime number known to humanity.
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This is the 15th prime number found by the Great Internet Mersenne Prime Search, or Gimps, for short, a volunteer project that has been running for 20 years.
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The next major goal for volunteers who join the search will be to find a prime number with 100 million digits — a goal which has a $150,000 prize.
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The number 5, for example, when viewed among the natural, or counting, numbers is one of those elemental creatures: a prime number, divisible only by 1 and itself.
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To this end, he developed what is arguably the smallest possible executable code (code that can be read by a machine) that can be represented by a prime number.
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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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You notice that the number 103 is a semi-prime number, and so you organize the symbols in a 3-by-5 grid and leave the O's as blank spaces.
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It's the first prime-number Super Bowl since XLVII (Ravens over 49ers, 34-31.) Among the athletes who have worn No. 53: Don Drysdale, Harry Carson, Artis Gilmore and Darryl Dawkins.
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It's the first prime-number Super Bowl since XLVII (Ravens over 25ers, 34-31.) Among the athletes who have worn No. 53: Don Drysdale, Harry Carson, Artis Gilmore and Darryl Dawkins.
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" (In case anyone asks, have the Amazon Prime number handy: $4.5 billion.) Finally, conclude with this bunker buster: "In 2017 there were 455 scripted series, and 2017 will end off the charts.
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In Sweden, Meshuggah, in the nineties, built roaring, ferocious songs atop fiendish riffs in prime-number time signatures; Opeth, in the aughts, found a connection between death-metal fury and Pink Floydian reverie.
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In the first entry of a cycle of videos titled beweistheorie I, Tarkhanov explores the symmetries between the language of art and the structure which is defined by a prime number sequence he discovered.
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Interestingly, the new prime number was actually discovered by the GIMPS machine on September 17th, 2015, but it took almost four months for a flesh-and-blood researcher to notice they actually had something.
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Within Windows, this is the natural home for writing old-school PC software in C++, C, and C#. In school, this meant making things like prime number checkers, record collection organizers, and Colossal Cave Adventure clones.
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In a Reddit "Ask Me Anything" Monday, Bill Gates solicited questions from the internet and answered dozens of them, from serious questions about the future of philanthropy to silly ones about his favorite prime number (it's 103).
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Assuming you can say two digits a second and have evolved beyond the need for food, sleep, or a social life, the largest prime number ever discovered would take you more than four months to even say.
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This obscenely large number, which you can peruse here (though it is a 44MB text document full of numbers) does little for the math community other than continuing the never-ending exploration of every possible prime number out there.
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It's an obvious statement—numbers are information—but one that might lead to absurd conclusions, as a computer scientist named Phil Carmody attempted to demonstrate in 2001 with the discovery and publication of a stupidly long prime number representing a section of forbidden computer code implementing a DVD decoding algorithm known as DeCSS.
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"In a video posted to YouTube by Crown Sterling, the company claims that it had identified "for the first time an infinitely predictable prime number pattern," as well as rolled out buzzwords ranging from "infinite wave conjugations" and "quasi prime numbers" to the "nano-scale of time" and "speed of AI oscillations.
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The nuts and bolts of producing a prime number that can represent a computer program are a bit messy (it involves padding the original program code), but the basic idea was that Carmody's prime would be so large that it would crack the top 20 largest known primes, and, as such, it would have to be listed publicly for that reason alone.
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It is a prime number, and therefore also disproves a conjecture of Richard K. Guy that the complexity of every prime number is one plus the complexity of ..
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263 is the natural number between 262 and 264. It is also a prime number.
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19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.
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The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. logarithmic. A prime number is a positive integer with no divisors other than 1 and itself, excluding 1. Euclid recorded a proof that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.
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367 is a prime number, Perrin number, self number, happy number, and a strictly non-palindromic number.
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97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number.
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A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 (25). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational.
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53 (fifty-three) is the natural number following 52 and preceding 54. It is the 16th prime number.
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This may, e.g., result from relevant knowledge which A possesses and B lacks, as in the case where one should decide the answer to the question 'Is 15 a prime number?' and A knows what a prime number is and B doesn't; the sentence then doesn't have a meaning for B.
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The prime number theorem for arithmetic progressions deals with the asymptotic distribution of prime numbers in an arithmetic progression.
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211 (two hundred [and] eleven) is the natural number between 210 and 212. It is also a prime number.
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149 (one hundred [and] forty-nine) is the natural number between 148 and 150. It is also a prime number.
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359 is a prime number, safe prime, Eisenstein prime with no imaginary part, Chen prime, and strictly non-palindromic number.
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The average order of the Möbius function is zero. This statement is, in fact, equivalent to the prime number theorem.
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269 (two hundred [and] sixty-nine) is the natural number between 268 and 270. It is also a prime number.
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227 (two hundred [and] twenty-seven) is the natural number between 226 and 228. It is also a prime number.
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241 (two hundred [and] forty-one) is the natural number between 240 and 242. It is also a prime number.
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251 (two hundred [and] fifty-one) is the natural number between 250 and 252. It is also a prime number.
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A megaprime is a prime number with at least one million decimal digitsChris Caldwell, The Prime Glossary: megaprime at The Prime Pages. Retrieved on 2008-01-04. (whereas titanic prime is a prime number with at least 1,000 digits, and gigantic prime has at least 10,000 digits). The number of megaprimes found by year.
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In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring. It states that every odd number exceeding 3 is either a prime number or the sum of three prime numbers. It follows from the generalized Riemann hypothesis, and (trivially) from Goldbach's weak conjecture.
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The unique factorization theorem indicates that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. For example: :90 = 2^1 \cdot 3^2 \cdot 5^1 = 2 \cdot 3 \cdot 3 \cdot 5. Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5.
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''' 127 (one hundred [and] twenty-seven) is the natural number following 126 and preceding 128. It is also a prime number.
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359 (three hundred [and] fifty-nine) is the natural number following 358 and preceding 360. 359 is the 72nd prime number.
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is .
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Often Eisenstein's criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer ) of for . The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift.
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103 is the smallest prime number in which the period length of its reciprocal is exactly 1/3 of the maximum length.
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Knapowski expanded on the work of others in several fields of number theory, prime number theorem, modular arithmetic and non-Euclidean geometry.
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In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
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Reo Fortune conjectured that no Fortunate number is composite (Fortune's conjecture). A Fortunate prime is a Fortunate number which is also a prime number.
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1 is by convention not considered a prime number; although universal today, this was a matter of some controversy until the mid-20th century.
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In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.
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The dimension of the Hilbert space is important when generating sets of mutually unbiased bases using Weyl groups. When d is a prime number, then the usual d + 1 mutually unbiased bases can be generated using Weyl groups. When d is not a prime number, then it is possible that the maximal number of mutually unbiased bases which can be generated using this method is 3.
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Bertrand's postulate, proven in 1852, states that there is always a prime number between k and 2k, so in particular pn+1 < 2pn, which means gn < pn. The prime number theorem, proven in 1896, says that the average length of the gap between a prime p and the next prime will asymptotically approach ln(p) for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem an upper bound on the length of prime gaps: For every \epsilon > 0, there is a number N such that for all n > N :g_n < p_n\epsilon.
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17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number. Seventeen is the sum of the first four prime numbers.
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In November 2018, MIT Press published two collections of articles from Quanta Magazine, Alice and Bob Meet the Wall of Fire and The Prime Number Conspiracy.
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Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits.
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Thus, when n+1 is prime, the first factor in the product becomes one, and the formula produces the prime number n+1. But when n+1 is not prime, the first factor becomes zero and the formula produces the prime number 2.. This formula is not an efficient way to generate prime numbers because evaluating n! \bmod (n+1) requires about n-1 multiplications and reductions \bmod (n+1).
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Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than 2^{16}.
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331 is a prime number, cuban prime, sum of five consecutive primes (59 + 61 + 67 + 71 + 73), centered pentagonal number, centered hexagonal number, and Mertens function returns 0.
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Cover of the first edition, published by Berkley Books. Prime Number published in 1970, is a collection of science fiction storiesHarry Harrison Bibliography by American writer Harry Harrison.
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In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
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The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (PNT) (Chandrasekharan, 1969).
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103 is the 27th prime number. The previous prime is 101, making them both twin primes. It is also a happy number. 103 is a strictly non-palindromic number.
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7 (seven) is the natural number following 6 and preceding 8. It is a prime number, and is often considered lucky in Western culture, and is often seen as highly symbolic.
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It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
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Other reasons include the sieve of Eratosthenes, and the definition of a prime number itself (a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.).
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On September 17, 2015 (but not actually noticed until January 7, 2016), Cooper discovered yet another Mersenne prime, 274,207,281 \- 1, which was the largest known prime number at 22,338,618 decimal digits.
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In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by . Selberg and Erdős both used this identity to give elementary proofs of the prime number theorem.
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However, is not a field because 12 is not a prime number. The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer , arithmetic "modulo " means to work with the numbers : The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a prime number.
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It's 6. multiply 0'3 by 6: 0'1 8 subtract: ———— 9' What times 3 ends in 9? It's 3. multiply 0'3 by 3: 0'9 subtract: ———— 9' repetition of earlier difference makes 3'6 two and two-thirds Now move the decimal point one place left, to get 3!6 four-fifteenths Removing common factors is annoying, and it is unnecessary if the base is a prime number. Computers use base 2, which is a prime number, so division always works.
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Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .
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In number theory, Rosser's theorem was published by J. Barkley Rosser in 1939. Its statement follows. Let pn be the nth prime number. Then for n ≥ 1 :p_n > n \cdot \ln n.
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Also, no untouchable number is three more than a prime number, except 5, since if p is an odd prime then the sum of the proper divisors of 2p is p + 3\.
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The characteristic of an integral domain is either 0 or a prime number. If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f(x) = xp is injective.
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A similar construction using the moment curve modulo a prime number, but in two dimensions rather than three, provides a linear bound for the no-three-in-line problem.Credited by to Paul Erdős.
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This article collects together a variety of proofs of Fermat's little theorem, which states that :a^p \equiv a \pmod p for every prime number p and every integer a (see modular arithmetic).
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A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer by Eratosthenes' method: # Create a list of consecutive integers from 2 through : . # Initially, let equal 2, the smallest prime number. # Enumerate the multiples of by counting in increments of from to , and mark them in the list (these will be ; the itself should not be marked).
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Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie–Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive. Any finite field F has the property that there is a unique prime number p such that px=0 for all x in F; that is, x added to itself p times is zero. For example, in F_2, the prime number has this property.
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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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43,112,609 is a prime number. Moreover, it is the exponent of the 47th Mersenne prime, equal to M43,112,609 = 243,112,609 − 1, a prime number with 12,978,189 decimal digits. It was discovered on August 23, 2008 by Edson Smith, a volunteer of the Great Internet Mersenne Prime Search. The 45th Mersenne prime, M37,156,667 = 237,156,667 − 1, was discovered two weeks later on September 6, 2008, marking the shortest chronological gap between discoveries of Mersenne primes since the formation of the online collaborative project in 1996.
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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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3 (three) is a number, numeral, and glyph. It is the natural number following 2 and preceding 4, and is the smallest odd prime number. It has religious or cultural significance in many societies.
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1093 is the natural number following 1092 and preceding 1094. 1093 is a prime number. Together with 1091 and 1097, it forms a prime triplet. It is a happy prime and a star prime.
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Let p be a fixed prime number. An infinite group G is called a Tarski Monster group for p if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has p elements.
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Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called smooth numbers and rough numbers, respectively.
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151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime. 151 is a centered decagonal number. 151 is also a lucky number.
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Then is finite. If is composite, is divisible by prime which is less than . From Cauchy's theorem, the subgroup will be exist whose order is , it is not suitable. Therefore, must be a prime number.
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139 is the 34th prime number. It is a twin prime with 137. Because 141 is a semiprime, 139 is a Chen prime. 139 is the smallest prime before a prime gap of length 10.
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The prime number has been used in this way. As Proth primes have simple binary representations, they have also been used in fast modular reduction without the need for pre-computation, for example by Microsoft.
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Vogel considers the theoretically infinite four-dimensional space of tones of his Tonnetz as complete; no further dimensions are needed for higher prime numbers. According to his theory, consonance results from the congruency of harmonics. The prime number 11 and any other higher prime number can not lead to any perception of congruency, as the inner ear separates only the first eight to ten partials. The eleventh partial may be audible and discriminable from the tenth or twelfth partial if isolated via techniques such as flageolet.
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He also studied, under the form of integer polynomials modulo both a prime number and an irreducible polynomial (remaining irreducible modulo that prime number), what can nowadays be recognized as finite fields (more general than those of prime order).David A. Cox, "Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first", American Mathematical Monthly 118 Vol 1, January 2011, pp. 3–31. See p. 10. He was educated in Königsberg and Berlin, where among his teachers were Jakob Steiner and Carl Gustav Jacob Jacobi.
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Wike's law of low odd primes is a methodological principle to help design sound experiments in psychology. It is: "If the number of experimental treatments is a low odd prime number, then the experimental design is unbalanced and partially confounded" (Wike, 1973, pp. 192-193). This law was stated by Edwin Wike in a humorous article in which he also admits that the association of his name with the law is an example of Stigler's law of eponymy. The lowest odd prime number is three.
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The answer they provide (either true or false) will be incorrect, or correct, with some bounded probability. For instance, the Solovay–Strassen primality test is used to determine whether a given number is a prime number. It always answers true for prime number inputs; for composite inputs, it answers false with probability at least and true with probability less than . Thus, false answers from the algorithm are certain to be correct, whereas the true answers remain uncertain; this is said to be a -correct false-biased algorithm.
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If the user is willing to tolerate an arbitrarily small chance that the number found is not a prime number but a pseudoprime, it is possible to use the much faster and simpler Fermat primality test.
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Simplicity: Simple for p a prime number. Order: p Schur multiplier: Trivial. Outer automorphism group: Cyclic of order p − 1\. Other names: Z/pZ, Cp Remarks: These are the only simple groups that are not perfect.
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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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11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name.
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The implausibility that a composite might have been unintentionally introduced where a prime number is required has led to the suspicion of sabotage to introduce a backdoor software vulnerability. This socat bug affected version 1.7.3.0 and 2.0.
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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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Euler genera are generated from the prime factors 3 and 5, whereas an Euler–Fokker genus can have factors of 7 or any higher prime number. The degree is the number of intervals which generate a genus.
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149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is an emirp, since the number 941 is also prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes. 149 is an irregular prime since it divides the numerator of the Bernoulli number B130. 149 is an Eisenstein prime with no imaginary part and a real part of the form 3n - 1.
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David Wilson Henderson showed, in 1963, that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number. He also showed that such symmetric Venn diagrams exist when n is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs, Killian, and Savage showed that symmetric Venn diagrams exist for all other primes. These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number.
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The distinction between elementary and non-elementary proofs has been considered especially important in regard to the prime number theorem. This theorem was first proved in 1896 by Jacques Hadamard and Charles Jean de la Vallée-Poussin using complex analysis. Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs: However, in 1948, Atle Selberg produced new methods which led him and Paul Erdős to find elementary proofs of the prime number theorem.
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An odd prime number p is defined to be regular if it does not divide the class number of the p-th cyclotomic field Q(ζp), where ζp is a primitive p-th root of unity, it is listed on . The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z(ζp) up to equivalence. Two ideals I,J are considered equivalent if there is a nonzero u in Q(ζp) so that I=uJ.
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For Russell, a denoting phrase is a semantically complex expression that can serve as the grammatical subject of a sentence. Paradigm examples include both definite descriptions ("the shortest spy") and indefinite descriptions ("some sophomore"). A phrase does not need to have a denotation to be a denoting phrase: "the greatest prime number" is a denoting phrase in Russell's sense even though there is no such thing as the greatest prime number. According to Russell's theory, denoting phrases do not contribute objects as the constituents of the singular propositions in which they occur.
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In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896.
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In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group M, which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31), as well as 41, 47, 59, and 71. The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. Supersingular primes are related to the notion of supersingular elliptic curves as follows.
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347 is a prime number, safe prime, Eisenstein prime with no imaginary part, Chen prime, Friedman number since 347 = 73 \+ 4, and a strictly non-palindromic number. It is the number of an area code in New York.
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379 is a prime number, Chen prime, and a happy number in base 10. It is the sum of the 15 consecutive primes (3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53).
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Localization and completion of a spectrum at a prime number p are both examples of Bousfield localization, resulting in a local spectrum. For example, localizing the sphere spectrum S at p, one obtains a local sphere S_{(p)}.
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Any prime number is clearly cyclic. All cyclic numbers are square-free.For if some prime square p2 divides n, then from the formula for φ it is clear that p is a common divisor of n and φ(n).
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Hilbert calls for a solution to the Riemann hypothesis, which has long been regarded as the deepest open problem in mathematics. Given the solution, he calls for more thorough investigation into Riemann's zeta function and the prime number theorem.
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Using the best published value for c at the time, an immediate consequence of his result was that :gn < pn5/8, where pn the n-th prime number and gn = pn+1 − pn denotes the n-th prime gap.
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373, prime number, balanced prime, sum of five consecutive primes (67 + 71 + 73 + 79 + 83), permutable prime with 337 and 733, palindromic prime in 3 consecutive bases: 5658 = 4549 = 37310 and also in base 4: 113114, two-sided primes.
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383, prime number, safe prime, Woodall prime, Thabit number, Eisenstein prime with no imaginary part, palindromic prime. It is also the first number where the sum of a prime and the reversal of the prime is also a prime.
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Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin (14 August 1866 – 2 March 1962) was a Belgian mathematician. He is best known for proving the prime number theorem. The king of Belgium ennobled him with the title of baron.
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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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On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number, , having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.
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5 (five) is a number, numeral, and glyph. It is the natural number following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand.
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In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
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Binary lambda calculus is designed from an algorithmic information theory perspective to allow for the densest possible code with the most minimal means, featuring a 29 byte self interpreter, a 21 byte prime number sieve, and a 112 byte Brainfuck interpreter.
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While the total supply ever available is 433.494.437 DRGN, which is a prime number. The Tokenized Micro-licenses are used throughout the platform and ecosystem. It is a new form of software licensing that can be applied in different ways.
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47 (forty-seven) is the natural number following 46 and preceding 48. It is a prime number, and appears in popular culture as the adopted favorite number of Pomona College and an obsession of the hip hop collective Pro Era.
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Titanic prime is a term coined by Samuel Yates in the 1980s, denoting a prime number of at least 1000 decimal digits. Few such primes were known then, but the required size is trivial for modern computers. The first 30 titanic primes are of the form: :p = 10^{999} + n, for n one of 7, 663, 2121, 2593, 3561, 4717, 5863, 9459, 11239, 14397, 17289, 18919, 19411, 21667, 25561, 26739, 27759, 28047, 28437, 28989, 35031, 41037, 41409, 41451, 43047, 43269, 43383, 50407, 51043, 52507 . The number of primes in this range is consistent with the expected number based on the prime number theorem.
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A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger.
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For a repdigit to be prime, it must be a repunit and have a prime number of digits in its base. In particular, as Brazilian repunits do not allow the number of digits to be exactly two, Brazilian primes must have an odd prime number of digits. Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 1114 = 3 × 7 and 111 = 11110 = 3 × 37 are not prime. In any given base b, every repunit prime in that base with the exception of 11b (if it is prime) is a Brazilian prime. The smallest Brazilian primes are :7 = 1112, 13 = 1113, 31 = 111112 = 1115, 43 = 1116, 73 = 1118, ... While the sum of the reciprocals of the prime numbers is a divergent series, the sum of the reciprocals of the Brazilian prime numbers is a convergent series whose value, called the "Brazilian primes constant", is slightly larger than 0.33 .
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The k-th homotopy group of a sphere spectrum is the k-th stable homotopy group of spheres. The localization of the sphere spectrum at a prime number p is called the local sphere at p and is denoted by S_{(p)}.
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The fact that there are two logarithms (log of a log) in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.
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Prime gap frequency distribution for primes up to 1.6 billion. Peaks occur at multiples of 6."Hidden structure in the randomness of the prime number sequence?", S. Ares & M. Castro, 2005 A prime gap is the difference between two successive prime numbers.
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107 is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime. Plugged into the equation 2^p - 1, 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime.
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Reiher received his Dr. rer. nat. from the University of Rostock under supervision of Hans-Dietrich Gronau in February 2010 (Thesis: A proof of the theorem according to which every prime number possesses property B) and works now at the University of Hamburg.
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In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field., pp. 27–28.
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2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
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Seventeen is the seventh prime number. The next prime is nineteen, with which it forms a twin prime. Seventeen is a permutable prime and a supersingular prime. Seventeen is the third Fermat prime, as it is of the form 2 + 1, specifically with n = 2.
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For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.
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6 (2002), pp. 409-424 If p is a prime number, then the p-rank of G is the largest rank of an elementary abelian p-subgroup. The sectional p-rank is the largest rank of an elementary abelian p-section (quotient of a subgroup).
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An Abelian simple group is either or cyclic group whose order is a prime number . Let is an Abelian group, then all subgroups of are normal subgroups. So, if is a simple group, has only normal subgroup that is either or . If , then is .
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The connected components of Field are the full subcategories of characteristic p, where p = 0 or is a prime number. Each such subcategory has an initial object: the prime field of characteristic p (which is Q if p = 0, otherwise the finite field Fp).
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Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number. The translations form a group of isometries. However, there is no pattern with this group as symmetry group.
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If the Riemann hypothesis is true, these fluctuations will be small, and the asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of x for intervals near a number x).
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1987 is an odd number and the 300th prime number. It is the first number of a sexy prime triplet (1987, 1993, 1999). Being of the form 4n + 3, it is a Gaussian prime. It is a lucky number and therefore also a lucky prime.
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For example, he proved the infinitude of primes using the divergence of the harmonic series, and used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.
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If q is congruent to 3 (mod 4) then −1 is not a square, and Q is a skew-symmetric matrix. When q is a prime number, Q is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation.
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Between the years 1965 and 1995 he worked at the Tata Institute of Fundamental Research and after retirement joined the National Institute of Advanced Studies, Bangalore where he worked till 2011, the year he died. During the course of his lifetime, he published over 200 articles, of which over 170 have been catalogued by Mathematical Reviews. His work was primarily in the area of prime number theory, working on the Riemann zeta function and allied functions. Apart from prime number theory, he made substantial contributions to the theory of transcendental number theory, in which he is known for his proof of the six exponentials theorem, achieved independently of Serge Lang.
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Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as : p = x^2 + y^2 with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Girard in 1625, and again by Fermat in 1640, but neither supplied a proof. The "only if" clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is congruent to 0, 1, or 2. An odd prime number is congruent to either 1 or 3 modulo 4, and the second possibility has just been ruled out.
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There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem. To state it precisely, let be the finite field with elements, for some fixed , and let be the number of monic irreducible polynomials over whose degree is equal to . That is, we are looking at polynomials with coefficients chosen from , which cannot be written as products of polynomials of smaller degree. In this setting, these polynomials play the role of the prime numbers, since all other monic polynomials are built up of products of them.
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The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the th term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.
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Most of Littlewood's work was in the field of mathematical analysis. He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship. However, the link between the Riemann hypothesis and the prime number theorem had been known before in Continental Europe, and Littlewood wrote later in his book, A Mathematician's Miscellany that his rediscovery of the result did not shed a positive light on the isolated nature of British mathematics at the time.
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In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.
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Fifty-nine is the 17th prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary part and real part of the form .
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Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number. It is an Eisenstein prime with no imaginary part and real part of the form . It is a Lucas number.
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Then we should be able to require the existence of n such that N − F(n) is both positive and a prime number; and with all the fi(n) prime numbers. Not many cases of these conjectures are known; but there is a detailed quantitative theory (Bateman–Horn conjecture).
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In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p-derivation is related to that of a derivation in differential algebra.
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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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Because 2 is a prime number, it must also divide p, by Euclid's lemma. So p = 2r, for some integer r. But then, :2q^2 = (2r)^2 = 4r^2, :q^2 = 2r^2, which shows that 2 must divide q as well. So q = 2s for some integer s.
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Fourth Estate) The Music of the Primes (British subtitle: Why an Unsolved Problem in Mathematics Matters; American subtitle: Searching to Solve the Greatest Mystery in Mathematics) is a 2003 book by Marcus du Sautoy, a professor in mathematics at the University of Oxford, on the history of prime number theory. In particular he examines the Riemann hypothesis, the proof of which would revolutionize our understanding of prime numbers. He traces the prime number theorem back through history, highlighting the work of some of the greatest mathematical minds along the way. The cover design for the hardback version of the book contains several pictorial depictions of prime numbers, such as the number 73 bus.
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Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area. Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square. The connected sum of two prime knots The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.
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In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses.
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In all dimensions, the inequality of arithmetic and geometric means shows that the volume of the pieces is less than the volume of the hypercube into which they should be packed. However, it is unknown whether the puzzle can be solved in five dimensions, or in higher prime number dimensions.
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In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in the paper .
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353 is a prime number, Chen prime, Proth prime, Eisenstein prime with no imaginary part, palindromic prime, and Mertens function returns 0. 353 is the base of the smallest 4th power that is the sum of 4 other 4th powers, discovered by Norrie in 1911: 3534 = 304 \+ 1204 \+ 2724 \+ 3154.
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In 1977, he proved that if p is an odd prime number, and the natural numbers x, y and z satisfy x^{2p} + y^{2p} = z^{2p}, then 2p must divide x or y.G. Terjanian, Sur l'equation x^{2p}+ y^{2p} = z^{2p} ', CR. Acad. Sc. Paris. ,. 285. (1977), 973-975.
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In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.
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Uniruledness behaves very differently in positive characteristic. In particular, there are uniruled (and even unirational) surfaces of general type: an example is the surface xp+1 \+ yp+1 \+ zp+1 \+ wp+1 = 0 in P3 over p, for any prime number p ≥ 5.T. Shioda, Math. Ann. 211 (1974), 233-236.
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Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving this. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem.
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The case k=2 can be proved by the Borsuk-Ulam theorem. When k is an odd prime number, the proof involves a generalization of the Borsuk-Ulam theorem. When k is a composite number, the proof is as follows (demonstrated for the measure-splitting variant). Suppose k=p\cdot q.
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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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In mathematics, specifically in the field of group theory, the McKay Conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number p to that of the normalizer of a Sylow p-subgroup. It is named after Canadian mathematician John McKay.
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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 strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e.
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The regular hendecagon has Dih11 symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z11, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and group order.
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317 is a prime number, Eisenstein prime with no imaginary part, Chen prime, and a strictly non-palindromic number. 317 is the exponent (and number of ones) in the fourth base-10 repunit prime.Guy, Richard; Unsolved Problems in Number Theory, p. 7 317 is also shorthand for the LM317 adjustable regulator chip.
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It only works with the OSCAR protocol and if both chat partners use Trillian. However, the key used for encryption is established using a Diffie–Hellman key exchange which only uses a 128 bit prime number as modulus, which is extremely insecure and can be broken within minutes on a standard PC.
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The difference between this and the diapason normal is due to confusion over the temperature at which the French standard should be measured. The initial standard was A = , but this was superseded by A = 440 Hz, possibly because 439 Hz was difficult to reproduce in a laboratory since 439 is a prime number.
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100 is a Harshad number in base 10, and also in base 4, and in that base it is a self-descriptive number. There are exactly 100 prime numbers whose digits are in strictly ascending order (e.g. 239, 2357 etc.). 100 is the smallest number whose common logarithm is a prime number (i.e.
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Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular number is a prime number. It is the largest happy number less than 500. There is no solution to the equation φ(x) = 496, making 496 a nontotient. E8 has real dimension 496.
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The regular heptadecagon has Dih17 symmetry, order 34. Since 17 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z17, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon. John Conway labels these by a letter and group order.
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When (Z/nZ)× is cyclic, its generators are called primitive roots modulo n. For a prime number p, the group (Z/pZ)× is always cyclic, consisting of the non-zero elements of the finite field of order p. More generally, every finite subgroup of the multiplicative group of any field is cyclic..
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The regular enneadecagon has Dih19 symmetry, order 38. Since 19 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z19, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon. John Conway labels these by a letter and group order.
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He further contributed significantly to the understanding of perfect numbers, which had fascinated mathematicians since Euclid. Euler made progress toward the prime number theorem and conjectured the law of quadratic reciprocity. The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way for Carl Friedrich Gauss.
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The regular tridecagon has Dih13 symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z13, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order.
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The regular pentagon has Dih5 symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z5, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order.
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At the time of its discovery, 2,147,483,647 was the largest known prime number. In 1811, Peter Barlow, not anticipating future interest in perfect numbers, wrote (in An Elementary Investigation of the Theory of Numbers): > Euler ascertained that 231 − 1 = 2147483647 is a prime number; and this is > the greatest at present known to be such, and, consequently, the last of the > above perfect numbers [i.e., 230(231 − 1)], which depends upon this, is the > greatest perfect number known at present, and probably the greatest that > ever will be discovered; for as they are merely curious, without being > useful, it is not likely that any person will attempt to find one beyond it. He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary.
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Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for were strong enough for him to prove Bertrand's postulate that there exists a prime number between and for any integer . An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, chiefly that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper that the idea to apply methods of complex analysis to the study of the real function originates.
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Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre- Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem. Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable.
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The smallest composite Mersenne number with prime exponent n is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. , 51 Mersenne primes are known. The largest known prime number, , is a Mersenne prime.
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That cipher can be regarded as a predecessor to the RSA (cryptosystem) since all that is needed to transform it into RSA is to change the arithmetic from modulo a prime number to modulo a composite number. In his spare time Stephen Pohlig was a keen kayaker known to many throughout the New England area.
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Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name., Section II.1, p.
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This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to .
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911 (nine hundred [and] eleven) is the integer following 910 and preceding 912. It is a prime number, a Sophie Germain prime and the sum of three consecutive primes (293 + 307 + 311). It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1\. Since 913 is a semiprime, 911 is a Chen prime.
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The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after Euclid and Leonhard Euler. It has been conjectured that there are infinitely many Mersenne primes.
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Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subject Classification for multiplicative number theory is 11Nxx.
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Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square). In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, .
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In 2008, Geek Pride Day was officially celebrated in the U.S., where it was heralded by numerous bloggers, coalescing around the launch of the Geek Pride Day website. Math author, Euler Book Prize winner, and geek blogger John Derbyshire announced that he would be appearing in the Fifth Avenue parade on the prime number float, dressed as number 57.
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This formula yields the decomposition for n = 101 in the table. Ahmes was suggested to have converted 2/p (where p was a prime number) by two methods, and three methods to convert 2/pq composite denominators.. Others have suggested only one method was used by Ahmes which used multiplicative factors similar to least common multiples.
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In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime number p. When p=2 these operations were introduced by and were named Pontrjagin squares by (with the term "Pontryagin square" also being used). They were generalized to arbitrary primes by .
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This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers of and the Betti numbers with coefficients in a field . These can differ, but only when the characteristic of is a prime number for which there is some -torsion in the homology.
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A second drawback is that it won't break up clustered keys. For example, the keys 123000, 456000, 789000, etc. modulo 1000 all map to the same address. This technique works well in practice because many key sets are sufficiently random already, and the probability that a key set will be cyclical by a large prime number is small.
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He became the Walker professor in 1970, and retired to become an emeritus professor in 1973. The Robert H. Breusch Prize in Mathematics, for the best senior thesis from an Amherst student, was endowed in his memory. As a mathematician, Breusch was known for his new proof of the prime number theorem. and for the many solutions he provided to problems posed in the American Mathematical Monthly. His thesis work combined Bertrand's postulate with Dirichlet's theorem on arithmetic progressions by showing that each of the progressions 3i + 1, 3i + 2, 4i + 1, and 4i + 3 (for i = 0, 1, 2, ...) contains a prime number between x and 2x for every x ≥ 7... For instance, he proved that for n > 47 there is at least one prime between n and (9/8)n.
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The first difference between the two algorithms is that Adler-32 sums are calculated modulo a prime number, whereas Fletcher sums are calculated modulo 24−1, 28−1, or 216−1 (depending on the number of bits used), which are all composite numbers. Using a prime number makes it possible for Adler-32 to catch differences in certain combinations of bytes that Fletcher is unable to detect. The second difference, which has the largest effect on the speed of the algorithm, is that the Adler sums are computed over 8-bit bytes rather than 16-bit words, resulting in twice the number of loop iterations. This results in the Adler-32 checksum taking between one-and-a-half to two times as long as Fletcher's checksum for 16-bit word aligned data.
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Vogel rejects the more obvious variant where prime number 2 is weighted with 2 because it leads to results that in his opinion do not comply with the perception of musically skilled listeners. Finally, the weighted sum is divided by the number of tones of the chord. This computation is done for both the higher and the lower reference tone.
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As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers. No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number. The only nontrivial square Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.
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Paul Stäckel. Paul Gustav Samuel Stäckel (20 August 1862, Berlin – 12 December 1919, Heidelberg) was a German mathematician, active in the areas of differential geometry, number theory, and non-Euclidean geometry. In the area of prime number theory, he used the term twin prime (in its German form, "Primzahlzwilling") for the first time. According to Tietze, in reference to the term Primzahlzwillinge (i.e.
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743 (seven hundred [and] forty three) is the natural number following 742 and preceding 744. It is a prime number. 743 is a Sophie Germain prime, because 2 × 743 + 1 = 1487 is also prime. There are exactly 743 independent sets in a four-dimensional (16 vertex) hypercube graph, and exactly 743 connected cubic graphs with 16 vertices and girth four.
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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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389, prime number, Eisenstein prime with no imaginary part, Chen prime, highly cototient number, self number, strictly non-palindromic number. Smallest conductor of a rank 2 Elliptic curve. Also, 389 equals the displacement in cubic inches of the famous Pontiac GTO V-8 engine of 1964–66. The port number for LDAP, and the name for the Fedora Directory Server project.
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Figurate numbers representing pentagons (including five) are called pentagonal numbers. Five is also a square pyramidal number. Five is the only prime number to end in the digit 5 because all other numbers written with a 5 in the ones place under the decimal system are multiples of five. As a consequence of this, 5 is in base 10 a 1-automorphic number.
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George Woltman, computer scientist and noted prime number hobbyist, in 1993. George Woltman (born November 10, 1957) is the founder of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project researching Mersenne prime numbers using his software Prime95. He graduated from the Massachusetts Institute of Technology (MIT) with a degree in computer science. He lives in North Carolina.
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First, an ordinary prime number (or rational prime) which is congruent to is also an Eisenstein prime. Second, 3 and any rational prime congruent to is equal to the norm of an Eisentein integer . Thus, such a prime may be factored as , and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.
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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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A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1\. The only known Wilson primes are 5, 13, and 563 ; if any others exist, they must be greater than 2.
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On 25 January 2005, she released seventh studio album Miho Komatsu 7 : prime number. The album includes four singles. On 3 April 2005, the on-air version of I just wanna hold you tight was broadcast on TV Tokyo as an ending theme for Anime television series MÄR. It became her anime theme song for the first time after six years.
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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 dependence on the field F is only through its characteristic. If the homology groups are torsion-free, the Betti numbers are independent of F. The connection of p-torsion and the Betti number for characteristic p, for p a prime number, is given in detail by the universal coefficient theorem (based on Tor functors, but in a simple case).
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An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(x) and π(x) with their asymptotic estimates change sign infinitely often. See p. 9 of the preprint version. In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, now known as Skewes' number.
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The set with an odd number of factors is just the primes between x1/2 and x, so by the prime number theorem its size is (1 + o(1)) x / ln x. Thus these sieve methods are unable to give a useful upper bound for the first set, and overestimate the upper bound on the second set by a factor of 2.
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In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, given a prime number p, where n is its index in the ordered set of prime numbers, . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, and so 17 is a strong prime. The first few strong primes are :11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 .
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In mathematics, a prime power is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. (The number 1 is not counted as a prime power.) The sequence of prime powers begins 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, ... . The prime powers are those positive integers that are divisible by exactly one prime number; prime powers are also called primary numbers, as in the primary decomposition.
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Skewes obtained a degree in civil engineering from the University of Cape Town before emigrating to England. He studied mathematics at Cambridge University and obtained a PhD in mathematics in 1938. He discovered the first Skewes's number in 1933. This is also referred to as the Riemann true Skewes's number owing to its relationship to the Riemann hypothesis as related to prime number theory.
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613 is a prime number, the first number of prime triplet (p, p + 4, p + 6), middle number of sexy prime triple (p − 6, p, p + 6). It is the index of a prime Lucas number.What's Special About This Number? 613 is a centered square number with 18 per side, a circular number of 21 with a square grid and 27 using a triangular grid.
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An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
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The small list of initial prime numbers constitute complete parameters for the algorithm to generate the remainder of the list. These generators are referred to as wheels. While each wheel may generate an infinite list of numbers, past a certain point the numbers cease to be mostly prime. The method may further be applied recursively as a prime number wheel sieve to generate more accurate wheels.
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Much definitive work on wheel factorization, sieves using wheel factorization, and wheel sieve, was done by Paul PritchardPritchard, Paul, "Linear prime-number sieves: a family tree," Sci. Comput. Programming 9:1 (1987), pp. 17–35.Paul Pritchard, A sublinear additive sieve for finding prime numbers, Communications of the ACM 24 (1981), 18–23. Paul Pritchard, Explaining the wheel sieve, Acta Informatica 17 (1982), 477–485.
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For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000. The average gap between primes increases as the natural logarithm of the integer, and therefore the ratio of the prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem.
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The vault is known as the "Tower of the Beast", located in a buried Martian city. It says that the key to opening it is 'factoring the ultimate prime number'. Brender does not believe the tale and the creature causes a stock market crash, bankrupting Brender to achieve its aim. Brender is forced by his circumstances to take a job as a space pilot.
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PrimeGrid is a volunteer distributed computing project which searches for very large (up to near-world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client.
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Infinite harmonic progressions are not summable (sum to infinity). It is not possible for a harmonic progression of distinct unit fractions (other than the trivial case where a = 1 and k = 0) to sum to an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.. As cited by .
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Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. the largest known prime number is a Mersenne prime with 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled.
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Prime gaps can be generalized to prime k-tuples, patterns in the differences between more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem., Prime k-tuples conjecture, pp. 201–202.
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Every such prime is the sum of a square and twice a square.Gauss, DA Art. 182 Gauss proved Let q = a2 \+ 2b2 ≡ 1 (mod 8) be a prime number. Then :2 is a biquadratic residue (mod q) if and only if a ≡ ±1 (mod 8), and :2 is a quadratic, but not a biquadratic, residue (mod q) if and only if a ≡ ±3 (mod 8).
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Modern computers compute division by methods that are faster than long division, with the more efficient ones relying on approximation techniques from numerical analysis. For division with remainder, see Division algorithm. In modular arithmetic (modulo a prime number) and for real numbers, nonzero numbers have a multiplicative inverse. In these cases, a division by may be computed as the product by the multiplicative inverse of .
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An Introduction to the Theory of Numbers is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. The book grew out of a series of lectures by Hardy and Wright and was first published in 1938. The third edition added an elementary proof of the prime number theorem, and the sixth edition added a chapter on elliptic curves.
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In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1\. This is an unsolved problem. It is known that φ(n) = n − 1 if and only if n is prime. So for every prime number n, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1\.
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Bateman & Diamond (2004) pp.334–335 (The second ~ is not part of the conjecture and is proven by integration by parts.) The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for π2(x) above.
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He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through the summation of this approximation function over the non- trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for \pi(x). Riemann knew of Pafnuty Chebyshev's work on the Prime Number Theorem. He had visited Dirichlet in 1852.
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Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Elements, Prop. IX.36). For example, the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number, as follows: :for p = 2: 21(22 − 1) = 2 × 3 = 6 :for p = 3: 22(23 − 1) = 4 × 7 = 28 :for p = 5: 24(25 − 1) = 16 × 31 = 496 :for p = 7: 26(27 − 1) = 64 × 127 = 8128. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number.
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In this acceptor, the only accepting state is state 7. A (possibly infinite) set of symbol sequences, called a formal language, is a regular language if there is some acceptor that accepts exactly that set. For example, the set of binary strings with an even number of zeroes is a regular language (cf. Fig. 5), while the set of all strings whose length is a prime number is not.
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'Belphegor's prime is the palindromic prime number ' (1030 \+ 666 × 1014 \+ 1), a number which reads the same both backwards and forwards and is only divisible by itself and one. It was discovered by Harvey Dubner. The name Belphegor refers to one of the Seven Princes of Hell, who was charged with helping people make ingenious inventions and discoveries. "Belphegor's prime" is a name coined by author Clifford A. Pickover.
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The tower floats off the surface of the planet without any detectable force or support holding it up. The puzzles cover most of mathematics, with various questions tackling triangular numbers, rotations of four-dimensional figures and their corresponding shadows, and arcane aspects of prime number theory. It is not known what the Spire guards, or why there should be so many puzzles. Disturbingly, the Spire also seems to be alive.
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In mathematics, specifically group theory, finite groups of prime power order p^n, for a fixed prime number p and varying integer exponents n\ge 0, are briefly called finite p-groups. The p-group generation algorithm by M. F. Newman and E. A. O'Brien is a recursive process for constructing the descendant tree of an assigned finite p-group which is taken as the root of the tree.
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Color wheel graph of . Black parts inside refer to numbers having large absolute values. The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).
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One can find several finite matrix fields of characteristic p for any given prime number p. In general, corresponding to each finite field there is a matrix field. Since any two finite fields of equal cardinality are isomorphic, the elements of a finite field can be represented by matrices. Contrary to the general case for matrix multiplication, multiplication is commutative in a matrix field (if the usual operations are used).
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Paul Pritchard, Fast compact prime number sieves (among others), Journal of Algorithms 4 (1983), 332–344. in formulating a series of different algorithms. To visualize the use of a factorization wheel, one may start by writing the natural numbers around circles as shown in the adjacent diagram. The number of spokes is chosen such that prime numbers will have a tendency to accumulate in a minority of the spokes.
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In Abhyankar's conjecture, S is fixed, and the question is what G can be. This is therefore a special type of inverse Galois problem. The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p. This is a normal subgroup, and the parameter n is defined as the minimum number of generators of :G/p(G).
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Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from the prime's square). Eratosthenes proposed a simple algorithm for finding prime numbers. This algorithm is known in mathematics as the Sieve of Eratosthenes. In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit.
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In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem.
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Consider a finite -group (that is, a group with order , where is a prime number and ). We are going to prove that every finite '-group has a non-trivial center. Since the order of any conjugacy class of must divide the order of , it follows that each conjugacy class that is not in the center also has order some power of , where . But then the class equation requires that .
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A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms.
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Primes p that divide 2 − 1, for some prime number n. 3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 () All Mersenne primes are, by definition, members of this sequence.
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Several very similar modern versions... of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which m is even or odd do not require separate arguments. It is sufficient to prove the theorem for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2).
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Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse.
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Fermat's little theorem states that if p is a prime number, then for any integer b, the number b − b is an integer multiple of p. Carmichael numbers are composite numbers which have this property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number, even though it is not actually prime.
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An illegal prime is a prime number that represents information whose possession or distribution is forbidden in some legal jurisdictions. One of the first illegal primes was found in 2001. When interpreted in a particular way, it describes a computer program that bypasses the digital rights management scheme used on DVDs. Distribution of such a program in the United States is illegal under the Digital Millennium Copyright Act.
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This result can be deduced from Fermat's little theorem, which states that . The base-10 repetend of the reciprocal of any prime number greater than 5 is divisible by 9.Gray, Alexander J., "Digital roots and reciprocals of primes", Mathematical Gazette 84.09, March 2000, 86. If the repetend length of for prime p is equal to p − 1 then the repetend, expressed as an integer, is called a cyclic number.
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In mathematics, a quartan prime is a prime number of the form x4 + y4, where x > 0, y > 0 (and x and y are integers). The odd quartan primes are of the form 16n + 1\. For example, 17 is the smallest odd quartan prime: 14 + 24 = 1 + 16 = 17\. With the exception of 2 (x = y = 1), one of x and y will be odd, and the other will be even.
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The prime numbers form a Behrend sequence, because every integer greater than one is a multiple of a prime number. More generally, a subsequence A of the prime numbers forms a Behrend sequence if and only if the sum of reciprocals of A diverges. The semiprimes, the products of two prime numbers, also form a Behrend sequence. The only integers that are not multiples of a semiprime are the prime powers.
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For example, before defining that the predicate "is a prime number", one first needs to define the collection of objects that might possibly satisfy the predicate, namely the set, N, of natural numbers. "Significance of the paradox" Irvine, A. D., "Russell's Paradox", The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N. Zalta (ed.) It functions as a formal definition of the function of meta-communication in communication.
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In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 -- a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes.
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Roger Alan Horn is an American mathematician specializing in matrix analysis. He was Research Professor of mathematics at the University of Utah. He is known for formulating the Bateman–Horn conjecture with Paul T. Bateman on the density of prime number values generated by systems of polynomials. His books Matrix Analysis and Topics in Matrix Analysis, co-written with Charles R. Johnson, are standard texts in advanced linear algebra.
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Though the index was not proposed as a serious method, it nevertheless has become popular in Internet discussions of whether a claim or an individual is cranky, particularly in physics (e.g., at the Usenet newsgroup sci.physics), or in mathematics. Chris Caldwell's Prime Pages has a version adapted to prime number research which is a field with many famous unsolved problems that are easy to understand for amateur mathematicians.
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150px Michele Cipolla (28 October 1880, Palermo - 7 September 1947, Palermo) was an Italian mathematician, mainly specializing in number theory. He was a professor of Algebraic Analysis at the University of Catania and, later, the University of Palermo. He developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the problem of binomial congruence.
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Cases of particular interest include those for which the space is a compact Riemann surface . The initial publication in 1956 of Atle Selberg dealt with this case, its Laplacian differential operator and its powers. The traces of powers of a Laplacian can be used to define the Selberg zeta function. The interest of this case was the analogy between the formula obtained, and the explicit formulae of prime number theory.
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In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley. The Paley construction uses quadratic residues in a finite field GF(q) where q is a power of an odd prime number. There are two versions of the construction depending on whether q is congruent to 1 or 3 (mod 4).
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In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) (for real values of the argument "s", as are works of Leonhard Euler, as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one. He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x. Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2\.
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A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside down), and surface (actual display or reflection on a mirror). The first few decimal dihedral primes are :2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 . The smallest dihedral prime that reads differently with each orientation and surface combination is 120121 which becomes 121021 (upside down), 151051 (mirrored), and 150151 (both upside down and mirrored). hex digits. The digits 0, 1 and 8 remain the same regardless of orientation or surface (the fact that 1 moves from the right to the left of the seven-segment cell when reversed is ignored). 2 and 5 remain the same when viewed upside down, and turn into each other when reflected in a mirror.
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Even and odd numbers: An integer is even if it is a multiple of two, and is odd otherwise. Prime number: An integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A number that can be factored into a product of smaller integers. Every integer greater than one is either prime or composite.
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Games are often played to 31, 61, or 121 points using a cribbage board to score. Primes is played similar to Fives and Threes except the only scoring plays are prime numbers. This generally keeps the games more competitive. For the bonus score at the end of the hand, the player who finished the hand receives points equal to tile with the most pips in competitors' hands, rounded down to the nearest prime number.
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Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write,Avigad, et al. (2007) discuss the process of formally verifying a proof of the prime number theorem.
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By the prime number theorem, the number of k-th powers of a prime below x is of the order x1/k/log x. From this, the number of t-term expressions with sums ≤x is roughly xt/k/(log x)t. It is reasonable to assume that for some sufficiently large number t this is x-c, i.e., all numbers up to x are t-fold sums of k-th powers of primes.
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Extending the ideas of Riemann, two proofs of the prime number theorem were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.
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However, it is strictly weaker. For example, is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as :6 = 2 \cdot 3 = (-2) \cdot (-3). In general, if is a unit, meaning a number with a multiplicative inverse in , and if is a prime element, then is also a prime element.
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In algebraic geometry the Sierpiński space arises as the spectrum, Spec(R), of a discrete valuation ring R such as Z(p) (the localization of the integers at the prime ideal generated by the prime number p). The generic point of Spec(R), coming from the zero ideal, corresponds to the open point 1, while the special point of Spec(R), coming from the unique maximal ideal, corresponds to the closed point 0.
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In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number p (which is suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. published the first account of the theories.
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Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory. Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered the first elementary proof for the prime number theorem, along with Atle Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg.
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Five is the third prime number. Because it can be written as , five is classified as a Fermat prime; therefore, a regular polygon with 5 sides (a regular pentagon) is constructible with compass and an unmarked straightedge. Five is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third factorial prime, also an alternating factorial.
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By assumption all coefficients in the product are divisible by p, leading to a contradiction. Therefore, the coefficients of the product can have no common divisor and are thus primitive. \square The proof is given below for the more general case. Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial in Z[X]; this explains the need for "non-constant" in the statement.
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More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper.
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These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive. (x), the prime counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers.
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This is the condition that it should be a subfield of where is a squarefree odd number. This result was introduced by in his Zahlbericht and by . In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take a prime number , has a normal integral basis consisting of all the -th roots of unity other than .
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Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space Zp of p-adic integers, where p is any prime number. Generalizing these examples, any product of finite discrete spaces is a Stone space, and the topological space underlying any profinite group is a Stone space. The Stone–Čech compactification of the natural numbers with the discrete topology, or indeed of any discrete space, is a Stone space.
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The Unix command time prints CPU time and elapsed real time for a Unix process. % gcc nextPrimeNumber.c -o nextPrimeNumber % time ./nextPrimeNumber 30000007 Prime number greater than 30000007 is 30000023 0.327u 0.010s 0:01.15 28.6% 0+0k 0+0io 0pf+0w This process took a total of 0.337 seconds of CPU time, out of which 0.327 seconds was spent in user space, and the final 0.010 seconds in kernel mode on behalf of the process.
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One of her professors was Derrick N. Lehmer, the number theorist well known for his work on prime number tables and factorizations. While working for him at Berkeley finding pseudosquares, she met his son, her future husband Derrick H. Lehmer. Upon her graduation summa cum laude with a B.A. in Mathematics (1928), Emma married the younger Lehmer. They moved to Brown University, where Emma received her M.Sc., and Derrick his Ph.D., both in 1930.
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3511 (three thousand, five hundred and eleven) is the natural number following 3510 and preceding 3512. 3511 is a prime number, and is also an emirp: a different prime when its digits are reversed. 3511 is a Wieferich prime, found to be so by N. G. W. H. Beeger in 1922 and the largest known – the only other being 1093. If any other Wieferich primes exist, they must be greater than 6.7.
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In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by . A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes. Beurling generalized the usual prime number theorem to Beurling generalized primes.
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When G is a finite abelian group, the group ring is commutative, and its structure is easy to express in terms of roots of unity. When R is a field of characteristic p, and the prime number p divides the order of the finite group G, then the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.
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Knapowski returned to Poznań to finish another thesis to complete a post-doctoral qualification needed to lecture at a German university. "On new "explicit formulas" in prime number theory" in 1960. In 1962 the Polish Mathematical Society awarded him their Mazurkiewicz Prize and he moved to Tulane University in New Orleans, United States. After a very short return to Poland, he left again and taught in Marburg in Germany, Gainesville, Florida and Miami, Florida.
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Although 57 is not prime, it is jokingly known as the "Grothendieck prime" after a story in which mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number. This story is repeated in Part 2 of a biographical article on Grothendieck in Notices of the American Mathematical Society. As a semiprime, 57 is a Blum integer since its two prime factors are both Gaussian primes. 57 is a 20-gonal number.
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Given a prime number and prime power with positive integers and such that , a primitive narrow-sense BCH code over the finite field (or Galois field) with code length and distance at least is constructed by the following method. Let be a primitive element of . For any positive integer , let be the minimal polynomial with coefficients in of . The generator polynomial of the BCH code is defined as the least common multiple .
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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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19 is a centered triangular number 19 is the 8th prime number, the seventh Mersenne prime exponent, and the second base-10 repunit prime exponent.Guy, Richard; Unsolved Problems in Number Theory, p. 7 19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k). In addition, 19 is a Heegner number and a centered hexagonal number.
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In algebra, more specifically group theory, a p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
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In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However, as the time progresses, many of these results have also been subsequently reproven using only elementary techniques.
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A computationally large safe prime is likely to be a cryptographically strong prime. Note that the criteria for determining if a pseudoprime is a strong pseudoprime is by congruences to powers of a base, not by inequality to the arithmetic mean of neighboring pseudoprimes. When a prime is equal to the mean of its neighboring primes, it's called a balanced prime. When it's less, it's called a weak prime (not to be confused with a weakly prime number).
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For finite groups, a "small neighborhood" is taken to be a subgroup defined in terms of a prime number p, usually the local subgroups, the normalizers of the nontrivial p-subgroups. In which case, a property is said to be local if it can be detected from the local subgroups. Global and local properties formed a significant portion of the early work on the classification of finite simple groups, which was carried out during the 1960s.
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Turán was very interested in the distribution of primes in arithmetic progressions, and he coined the term "prime number race" for irregularities in the distribution of prime numbers among residue classes. With his coauthor Knapowski he proved results concerning Chebyshev's bias. The Erdős–Turán conjecture makes a statement about primes in arithmetic progression. Much of Turán's number theory work dealt with the Riemann hypothesis and he developed the power sum method (see below) to help with this.
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This (along with much else) led to quantitative progress on Waring's problem, as part of the Hardy–Littlewood circle method, as it became known. In prime number theory, they proved results and some notable conditional results. This was a major factor in the development of number theory as a system of conjectures; examples are the first and second Hardy–Littlewood conjectures. Hardy's collaboration with Littlewood is among the most successful and famous collaborations in mathematical history.
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The sieve of Eratosthenes can be expressed in pseudocode, as follows:, p. 16.Jonathan Sorenson, An Introduction to Prime Number Sieves, Computer Sciences Technical Report #909, Department of Computer Sciences University of Wisconsin-Madison, January 2, 1990 (the use of optimization of starting from squares, and thus using only the numbers whose square is below the upper limit, is shown). algorithm Sieve of Eratosthenes is input: an integer n > 1. output: all prime numbers from 2 through n.
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Behrend's work covered a wide range of topics, and often consisted of "a new approach to questions already deeply studied". He began his research career in number theory, publishing three papers by the age of 23. His doctoral work provided upper and lower bounds on the density of the abundant numbers. He also provided elementary bounds on the prime number theorem, before that problem was solved more completely by Paul Erdős and Atle Selberg in the late 1940s.
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Ore is known for his work in ring theory, Galois connections, and most of all, graph theory. His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He then worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring. He then examined polynomial rings over skew fields, and attempted to extend his work on factorisation to non-commutative rings.
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Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression :a, a+q, a+2q, a+3q, \dots can have more than one prime only when its remainder a and modulus q are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.
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In the theory of finite groups the Sylow theorems imply that, if a power of a prime number p^n divides the order of a group, then the group has a subgroup of order p^n. By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable. For the Sylow theorems see p. 43; for Lagrange's theorem, see p.
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229 (two hundred [and] twenty-nine) is the natural number following 228 and preceding 230. It is a prime number, and a regular prime. It is also a full reptend prime, meaning that the decimal expansion of the unit fraction 1/229 repeats periodically with as long a period as possible. With 227 it is the larger of a pair of twin primes, and it is also the start of a sequence of three consecutive squarefree numbers.
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If gcd(a, b) = 1, then a and b are said to be coprime (or relatively prime). This property does not imply that a or b are themselves prime numbers. For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7\. Nevertheless, 6 and 35 are coprime. No natural number other than 1 divides both 6 and 35, since they have no prime factors in common.
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A prime number, often shortened to just prime, is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered.
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An Adler-32 checksum is obtained by calculating two 16-bit checksums A and B and concatenating their bits into a 32-bit integer. A is the sum of all bytes in the stream plus one, and B is the sum of the individual values of A from each step. At the beginning of an Adler-32 run, A is initialized to 1, B to 0. The sums are done modulo 65521 (the largest prime number smaller than 216).
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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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In 1936, he proved Rosser's trick, a stronger version of Gödel's first incompleteness theorem, showing that the requirement for ω-consistency may be weakened to consistency. Rather than using the liar paradox sentence equivalent to "I am not provable," he used a sentence that stated "For every proof of me, there is a shorter proof of my negation". In prime number theory, he proved Rosser's theorem. The Kleene–Rosser paradox showed that the original lambda calculus was inconsistent.
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Glyn Harman (born 2 November 1956) is a British mathematician working in analytic number theory. One of his major interests is prime number theory. He is best known for results on gaps between primes and the greatest prime factor of p + a, as well as his lower bound for the number of Carmichael numbers up to X. His monograph Prime-detecting Sieves (2007) was published by Princeton University Press. He has also written a book Metric Number Theory (1998).
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It has been conjectured that there are infinitely many decimal repunit primes.Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages The binary repunits are the Mersenne numbers and the binary repunit primes are the Mersenne primes. It is unknown whether there are infinitely many Brazilian primes. If the Bateman–Horn conjecture is true, then for every prime number of digits there would exist infinitely many repunit primes with that number of digits (and consequentially infinitely many Brazilian primes).
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If the sign of the offset is alternated (e.g. +1, −4, +9, −16, etc.), and if the number of buckets is a prime number p congruent to 3 modulo 4 (e.g. 3, 7, 11, 19, 23, 31, etc.), then the first p offsets will be unique (modulo p). In other words, a permutation of 0 through p-1 is obtained, and, consequently, a free bucket will always be found as long as at least one exists.
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Equivalently, any division algebra of period dividing n is Brauer equivalent to a tensor product of cyclic algebras of degree n. Even for a prime number p, there are examples showing that a division algebra of period p need not be actually isomorphic to a tensor product of cyclic algebras of degree p.Gille & Szamuely (2006), Remark 2.5.8. It is a major open problem (raised by Albert) whether every division algebra of prime degree over a field is cyclic.
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All other capabilities were the same in both models - RPN expression logic, 98 program memory locations, statistical functions, and 30 registers. Users could develop software for the HP-29C/19C, such as a prime number generator. The calculators expanded the HP-25's program capabilities by adding subroutines, increment/decrement looping, relative branching and indirect addressing (via register 0 as index). HP's internal code name for the 29C was Bonnie, the 19C was correspondingly named Clyde.
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The distribution needs to be uniform only for table sizes that occur in the application. In particular, if one uses dynamic resizing with exact doubling and halving of the table size, then the hash function needs to be uniform only when the size is a power of two. Here the index can be computed as some range of bits of the hash function. On the other hand, some hashing algorithms prefer to have the size be a prime number.
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The special case , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n, so only prime values of n need further investigation.If the exponent n were not prime or 4, then it would be possible to write n either as a product of two smaller integers (n = PQ), in which P is a prime number greater than 2, and then an = aPQ = (aQ)P for each of a, b, and c. That is, an equivalent solution would also have to exist for the prime power P that is smaller than n; or else as n would be a power of 2 greater than 4, and writing n = 4Q, the same argument would hold. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes.
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The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials ƒ1, ..., ƒm with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if there were such a prime p, having all values of the polynomials simultaneously prime for a given n would imply that at least one of them must be equal to p, which can only happen for finitely many values of n or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many n. An integer n is prime-generating for the given system of polynomials if every polynomial ƒi(n) produces a prime number when given n as its argument.
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Proofs of the prime number theorem before 1928 and only using the behaviour of the zeta function on the line Re s = 1 (as the 1908 proof of Edmund Landau), also appealed to some bound on the order of growth of the zeta function on this line. Returning to Japan after studying with Dr Wiener, he taught at Osaka University and the Tokyo Institute of Technology. He translated Cybernetics: Or Control and Communication in the Animal and Machine into Japanese.Wiener, N. (1956).
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If presented as part of a complex tone comprising also the adjoining partials these partials would fuse together, and a congruency with the partial of another note could no longer be detected. It might be interesting to test whether it is possible to detect harmonic congruency for higher prime numbers for instruments with odd partials, as the distances between partials are higher in these instruments. With training it might be possible to detect consonance up to prime number 17 or even 19.
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We then deduce from this knowledge that there is a prime number greater than two. Thus, it can be said that intuition and deduction combined to provide us with a priori knowledge – we gained this knowledge independently of sense experience. Empiricists such as David Hume have been willing to accept this thesis for describing the relationships among our own concepts. In this sense, empiricists argue that we are allowed to intuit and deduce truths from knowledge that has been obtained a posteriori.
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A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin (2004), and various wheel sieves are most common. A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left.
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As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods. Today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a sub-base system, is sexagesimal, base 60, which used 10 as a sub-base. Each quinary digit has log25 (approx.
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The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. Because the modulus is prime, Lagrange's theorem applies: a polynomial of degree can only have at most roots. In particular, has at most 2 solutions for each . This immediately implies that besides 0, there are at least distinct quadratic residues modulo : each of the possible values of can only be accompanied by one other to give the same residue.
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Circle of fifths in 19 tone equal temperament Major chord on C in 19 equal temperament: All notes within 8 cents of just intonation (rather than 14 for 12 equal temperament). , , or . Because 19 is a prime number, repeating any fixed interval in this tuning system cycles through all possible notes; just as one may cycle through 12 EDO on the circle of fifths, since a fifth is 7 semitones, and number 7 does not divide 12 evenly (7 is coprime to 12).
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After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents.Ribenboim, pp. 1–2. In other words, it was necessary to prove only that the equation an \+ bn = cn has no positive integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the factors of n.
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For illustration, let n be factored into d and e, n = de. The general equation : an \+ bn = cn implies that (ad, bd, cd) is a solution for the exponent e : (ad)e \+ (bd)e = (cd)e. Thus, to prove that Fermat's equation has no solutions for n > 2, it would suffice to prove that it has no solutions for at least one prime factor of every n. Each integer n > 2 is divisible by 4 or by an odd prime number (or both).
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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 same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes. ; deep:A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found.
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The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis HN) does. That requires an extra polynomial F(x) , which in the Goldbach problem would just be x , for which :N − F(n) is required to be a prime number, also. This is cited in Halberstam and Richert, Sieve Methods. The conjecture here takes the form of a statement when N is sufficiently large, and subject to the condition :Q(n)(N − F(n)) has no fixed divisor > 1\.
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In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography. Contrast with probable prime, which is likely (but not certain) to be prime, based on the output of a probabilistic primality test. In principle, every prime number can be proved to be prime in polynomial time by using the AKS primality test.
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Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127 (these are listed in Sloane's ). The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol emirp.
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Inuit counting also has sub-bases at 5, 10, and 15. Also called quinary (base- or pental) this is a numeral system with five as the base. The Kaktovik numerals take these sub-bases into account, as the base of the symbols change after 4, 9 and 14. As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.
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Interviewed Newman's sister He was an avid problem-solver, and as an undergraduate was a Putnam Fellow all three years he took part in the Putnam math competition; only the third person to attain that feat.See Joseph Gallian's history of the competition and the official MAA record His mathematical specialties included complex analysis, approximation theory and number theory. In 1980 he found a short proof of the prime number theorem, which can now be found in his textbook on Complex analysis.
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Nicolas, "Répartition des nombres superabondants", Bull. Math. Soc. France 103 (1975), pp. 65–90. In their 1944 paper, Alaoglu and Erdős conjectured that the ratio of two consecutive colossally abundant numbers was always a prime number. They showed that this would follow from a special case of the four exponentials conjecture in transcendental number theory, specifically that for any two distinct prime numbers p and q, the only real numbers t for which both pt and qt are rational are the positive integers.
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109 is the 29th prime number, so it is a prime with a prime subscript. The previous prime is 107, making them both twin primes. 109 is a centered triangular number. There are exactly 109 different families of subsets of a three-element set whose union includes all three elements, 109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements, and 109 squares on an infinite chessboard that can be reached by a knight within three moves.
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If F(X) = a_0 + a_1 X + \dots + a_n X^n is a polynomial with integer coefficients, then F is called primitive if the greatest common divisor of all the coefficients a_0, a_1, \dots, a_n is 1; in other words, no prime number divides all the coefficients. Proof: Clearly the product f(x).g(x) of two primitive polynomials has integer coefficients. Therefore, if it is not primitive, there must be a prime p which is a common divisor of all its coefficients.
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As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic. For any field F, there is a minimal subfield, namely the ', the smallest subfield containing 1F. It is isomorphic either to the rational number field Q', or to a finite field of prime order, Fp; the structure of the prime field and the characteristic each determine the other.
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Several cases of the conjecture have been proven to be true—for instance, it is known to be true for graphs with a prime number of vertices.—but the full conjecture remains open. Variants of the problem for randomized algorithms and quantum algorithms have also been studied. Bender and Ron have shown that, in the same model used for the evasiveness conjecture, it is possible in only constant time to distinguish directed acyclic graphs from graphs that are very far from being acyclic.
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In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g. The rank of the Hasse–Witt matrix is the Hasse or Hasse–Witt invariant.
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It managed to produce by itself the notion of prime number and the Goldbach conjecture. As with Racter, the question is how much the programmer filtered the output of the program, keeping only the occasional interesting output. Also, mathematics being a very specialized domain, it is doubtful whether the techniques used can be abstracted to general cognition. Another mathematical program, called Geometry, was celebrated for making an insightful discovery of an original proof that an isosceles triangle has equal base angles.
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The precise definition of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. Unlike "even", some mathematical terms are purposefully constructed to exclude trivial or degenerate cases. Prime numbers are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach, Lambert, Legendre, Cayley, and Kronecker wrote that 1 was prime. The modern definition of "prime number" is "positive integer with exactly 2 factors", so 1 is not prime.
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The Pythagorean prime 5 and its square root are both hypotenuses of right triangles with integer legs. The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse. A Pythagorean prime is a prime number of the form 4n + 1\. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares.
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The Adler-32 checksum is a specialization of the Fletcher-32 checksum devised by Mark Adler. The modulus selected (for both sums) is the prime number 65,521 (65,535 is divisible by 3, 5, 17 and 257). The first sum also begins with the value 1. The selection of a prime modulus results in improved "mixing" (error patterns are detected with more uniform probability, improving the probability that the least detectable patterns will be detected, which tends to dominate overall performance).
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Mathematics Made Difficult is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs. Although the aim is largely satirical,, page 6. it also shows the non-trivial mathematics behind operations normally considered obvious, such as numbering, counting, and factoring integers. As an example, the proof that 2 is a prime number starts: > It is easily seen that the only numbers between 0 and 2, including 0 but > excluding 2, are 0 and 1.
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The paradigmatic example of folding by characters is to add up the integer values of all the characters in the string. A better idea is to multiply the hash total by a constant, typically a sizeable prime number, before adding in the next character, ignoring overflow. Using exclusive 'or' instead of add is also a plausible alternative. The final operation would be a modulo, mask, or other function to reduce the word value to an index the size of the table.
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400: "1er Théorème. Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières … " (1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways … ) The case k = 1 of de Polignac's conjecture is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the prime number theorem.
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Usually, a text based on natural numbers (for example, the article "prime number") does not specify the used definition of natural numbers. Likewise, a text based on topological spaces (for example, the article "homotopy", or "inductive dimension") does not specify the used definition of a topological space. Thus, it is possible (and rather probable) that the reader and the author interpret the text differently, according to different definitions. Nevertheless, the communication is successful, which means that such different definitions may be thought of as equivalent.
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In fact, if is a divisor of such that , then is a divisor of such that . If one tests the values of in increasing order, the first divisor that is found is necessarily a prime number, and the cofactor cannot have any divisor smaller than . For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of that is not smaller than and not greater than . There is no need to test all values of for applying the method.
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The following protocol was suggested by David Chaum. A group, G, is chosen in which the discrete logarithm problem is intractable, and all operation in the scheme take place in this group. Commonly, this will be the finite cyclic group of order p contained in Z/nZ, with p being a large prime number; this group is equipped with the group operation of integer multiplication modulo n. An arbitrary primitive element (or generator), g, of G is chosen; computed powers of g then combine obeying fixed axioms.
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It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so P(1) \equiv 0 \pmod p. But for any integer k, P(1+kp) \equiv 0 \pmod p also, so P(1+kp) cannot also be prime (as it would be divisible by p) unless it were p itself.
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A prime number q is a strong prime if and both have some large (around 500 digits) prime factors. For a safe prime , the number naturally has a large prime factor, namely p, and so a safe prime q meets part of the criteria for being a strong prime. The running times of some methods of factoring a number with q as a prime factor depend partly on the size of the prime factors of . This is true, for instance, of the p−1 method.
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Safe primes are also important in cryptography because of their use in discrete logarithm-based techniques like Diffie–Hellman key exchange. If is a safe prime, the multiplicative group of numbers modulo has a subgroup of large prime order. It is usually this prime- order subgroup that is desirable, and the reason for using safe primes is so that the modulus is as small as possible relative to p. A prime number p = 2q + 1 is called a safe prime if q is prime.
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The characteristic set of a linear matroid is defined as the set of characteristics of the fields over which it is linear. For every prime number p there exist infinitely many matroids whose characteristic set is the singleton set {p},. and for every finite set of prime numbers there exists a matroid whose characteristic set is the given finite set.. If the characteristic set of a matroid is infinite, it contains zero; and if it contains zero then it contains all but finitely many primes., p. 225.
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In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary.
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In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: : 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... .
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Euler showed that \zeta(2)=\pi^2/6., Chapter 35, Estimating the Basel problem, pp. 205–208. The reciprocal of this number, 6/\pi^2, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common). The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the n-th prime is known.
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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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The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots. The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number . A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue.
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This prime number is called the characteristic of the field. Suppose that F is a field of characteristic p, and consider the function f(x) = x^p that raises each element of F to the power p. This is called the Frobenius automorphism of F. It is an automorphism of the field because of the Freshman's dream identity (x+y)^p = x^p+y^p. The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield.
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223 (two hundred [and] twenty-three) is the natural number between 222 and 224. 223 is a prime number. It is the smallest prime for which the two nearest primes on either side of it are 16 units apart. Among the 720 permutations of the numbers from 1 to 6, exactly 223 of them have the property that at least one of the numbers is fixed in place by the permutation and the numbers less than it and greater than it are separately permuted among themselves.
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No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors (this situation happens at the case for 28). Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, all Mersenne numbers are not untouchable, since Mn=2n-1 can be expressed as 2n's proper divisors' sum. No untouchable number is one more than a prime number, since if p is prime, then the sum of the proper divisors of p2 is p + 1\.
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The Gaussian coefficients count subspaces of a finite vector space. Let q be the number of elements in a finite field. (The number q is then a power of a prime number, , so using the letter q is especially appropriate.) Then the number of k-dimensional subspaces of the n-dimensional vector space over the q-element field equals : \binom nk_q . Letting q approach 1, we get the binomial coefficient : \binom nk, or in other words, the number of k-element subsets of an n-element set.
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257 is a prime number of the form 2^{2^n}+1, specifically with n = 3, and therefore a Fermat prime. Thus a regular polygon with 257 sides is constructible with compass and unmarked straightedge. It is currently the second largest known Fermat prime.. It is also a balanced prime, an irregular prime, a prime that is one more than a square, and a Jacobsthal–Lucas number. There are exactly 257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes).
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"PRIMES is in P" was a breakthrough in theoretical computer science. This article, published by Manindra Agrawal, Nitin Saxena, and Neeraj Kayal in August 2002, proves that the famous problem of checking primality of a number can be solved deterministically in polynomial time. The authors received the 2006 Gödel Prize and 2006 Fulkerson Prize for this work. Because primality testing can now be done deterministically in polynomial time using the AKS primality test, a prime number could itself be considered a certificate of its own primality.
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Cameron moved to the University of Göttingen, in Germany, to take two more semesters of math, and finally, she enrolled at the University of Marburg, for three semesters. Under the supervision of distinguished mathematician Kurt Hensel, Cameron wrote her dissertation On the decomposition of a prime number in a composed body. Before her degree was officially completed, however, there was one additional barrier for her to surmount. It seems she had completed the work and won the approval of her advisor, Dr. Hensel, without realizing a lesser-known caveat for graduation from a German university.
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Seventeen is the minimum number of vertices on a graph such that, if the edges are coloured with three different colours, there is bound to be a monochromatic triangle. (See Ramsey's theorem.) Seventeen is the only prime number which is the sum of four consecutive primes (2,3,5,7). Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime. The sequence of residues (mod n) of a googol and googolplex, for n = 1, 2, 3, ..., agree up until n = 17.
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An important conjecture due to Catalan, sometimes called the Catalan–Dickson conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers. The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the Lehmer five (named after D.H. Lehmer): 276, 552, 564, 660, and 966.
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A finite plane of order n is one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane). One major open question in finite geometry is: :Is the order of a finite plane always a prime power? This is conjectured to be true. Affine and projective planes of order n exist whenever n is a prime power (a prime number raised to a positive integer exponent), by using affine and projective planes over the finite field with elements.
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53, 2004 Later, Riemann considered this function for complex values of s and showed that this function can be extended to a meromorphic function on the entire plane with a simple pole at s = 1\. This function is now known as the Riemann Zeta function and is denoted by ζ(s). There is a plethora of literature on this function and the function is a special case of the more general Dirichlet L-functions. Analytic number theorists are often interested in the error of approximations such as the prime number theorem.
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A compound expression might be in the form EasilyComputed or LotsOfWork so that if the easy part gives true a lot of work could be avoided. For instance, suppose a large number N is to be checked to determine if it is a prime number and a function IsPrime(N) is available, but alas, it can require a lot of computation to evaluate. Perhaps N=2 or [Mod(N,2)≠0 and IsPrime(N)] will help if there are to be many evaluations with arbitrary values for N.
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A number that belongs to a singleton club, because no other number is "friendly" with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary . For a prime number p we have σ(p) = p + 1, which is co-prime with p.
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An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversible prime is used to mean the same as emirp, but may also, ambiguously, include the palindromic primes. The sequence of emirps begins 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, ... .
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Reduction of an abelian variety A modulo a prime ideal of (the integers of) K -- say, a prime number p -- to get an abelian variety Ap over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory. Here a refined theory of (in effect) a right adjoint to reduction mod p -- the Néron model -- cannot always be avoided.
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Hoheisel is known for a result on gaps between prime numbers:G. Hoheisel, Primzahlprobleme in der Analysis, Berliner Sitzungsberichte, pages 580-588, (1930) He proved that if π(x) denotes the prime-counting function, then there exists a constant θ < 1 such that :π(x + xθ) − π(x) ~ xθ/log(x), as x tends to infinity, implying that if pn denotes the n-th prime number then :pn+1 − pn < pnθ, for all sufficiently large n. In fact he showed that one may take :θ = 32999/33000 = 1 - 0.000(03), with (03) denoting periodic repeatition.
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In mathematics, specifically group theory, Cauchy's theorem states that if is a finite group and is a prime number dividing the order of (the number of elements in ), then contains an element of order . That is, there is in such that is the smallest positive integer with = , where is the identity element of . It is named after Augustin-Louis Cauchy, who discovered it in 1845. The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of .
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Although little is known about the life of Thymaridas, it is believed that he was a rich man who fell into poverty. It is said that Thestor of Poseidonia traveled to Paros in order to help Thymaridas with the money that was collected for him. Iamblichus states that Thymaridas called prime numbers "rectilinear", since they can only be represented on a one-dimensional line. Non-prime numbers, on the other hand, can be represented on a two-dimensional plane as a rectangle with sides that, when multiplied, produce the non-prime number in question.
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In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the prime number theorem. To be specific, we let π(x) denote the number of closed geodesics whose norm (a function related to length) is less than or equal to x; then π(x) ∼ x/ln(x). This result is usually credited to Atle Selberg. In his 1970 Ph.D. thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem.
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This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons: :A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none). (A Fermat prime is a prime number of the form 2^{(2^n)}+1.) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.
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In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article. A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime.
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In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, n! \equiv -1 \mod p but p ot\equiv 1 \mod n. The first few Pillai primes are :23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers.
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In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by . Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator.
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In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime- counting function and is the natural logarithm of .
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In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or a finite extension of the field of formal Laurent series Fq((T)) over a finite field Fq.
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In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.
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The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization: :252 = 2 × 3 × 7 Euclid's Elements first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss. The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number.
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Separable polynomials are used to define separable extensions: A field extension is a separable extension if and only if for every , which is algebraic over , the minimal polynomial of over is a separable polynomial. Inseparable extensions (that is extensions which are not separable) may occur only in characteristic . The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X) = 0. Thus we must have :P(X) = Q(Xp) for some polynomial Q over K, where the prime number p is the characteristic.
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Every cycle graph is a circulant graph, as is every crown graph with vertices. The Paley graphs of order (where is a prime number congruent to ) is a graph in which the vertices are the numbers from 0 to and two vertices are adjacent if their difference is a quadratic residue modulo . Since the presence or absence of an edge depends only on the difference modulo of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph.
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In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by : J(\chi,\psi) = \sum \chi(a) \psi(1 - a) \,, where the summation runs over all residues (for which neither a nor is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy.
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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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Adi Shamir, co-inventor of RSA (the others are Ron Rivest and Leonard Adleman) The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.
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A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units. A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of commutative algebra: it is possible for an irreducible quaternion to divide a product ab without dividing either a or b. Every Hurwitz quaternion can be factored as a product of irreducible quaternions.
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Since the actual math is performed in GF((2m)n), the reducing polynomial for GF((2m)n) is usually primitive and β = x in GF((2m)n). In order to meet the compatibility constraint for addition and multiplication, a search is done to choose any primitive element α of GF(2k) that will meet the constraint. Mapping to a composite field can be generalized to map GF(pk) to a composite field such as GF((pm)n), for p a prime number greater than 2, but such fields are not commonly used in practice.
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Isabelle has been used to formalize numerous theorems from mathematics and computer science, like Gödel's completeness theorem, Gödel's theorem about the consistency of the axiom of choice, the prime number theorem, correctness of security protocols, and properties of programming language semantics. Many of the formal proofs are maintained in the Archive of Formal Proofs, which contains (as of 2019) at least 500 articles with over 2 million lines of proof in total. The Isabelle theorem prover is free software, released under the revised BSD license. Isabelle was named by Lawrence Paulson after Gérard Huet's daughter.
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There is no formula to calculate prime numbers. However, the distribution of primes can be statistically modelled. The prime number theorem, which was proven at the end of the 19th century, says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits (logarithm). At the start of the 19th century, Adrien- Marie Legendre and Carl Friedrich Gauss suggested that as x goes very large, the number of primes up to x is asymptotic to x/\log x, where \log x is the natural logarithm of x.
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If G is cyclic then the transfer takes any element y of G to y[G:H]. A simple case is that seen in the Gauss lemma on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number p, with respect to the subgroup {1, −1}. One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that p − 1 is divisible by three.
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In the branch of mathematics known as additive combinatorics, Kneser's theorem can refer to one of several related theorems regarding the sizes of certain sumsets in abelian groups. These are named after Martin Kneser, who published them in 1953 and 1956. They may be regarded as extensions of the Cauchy–Davenport theorem, which also concerns sumsets in groups but is restricted to groups whose order is a prime number. The first three statements deal with sumsets whose size (in various senses) is strictly smaller than the sum of the size of the summands.
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A negative claim is a colloquialism for an affirmative claim that asserts the non-existence or exclusion of something. Claiming that it is impossible to prove a negative is a pseudologic, because there are many proofs that substantiate negative claims in mathematics, science, and economics, including Euclid's theorem, which proves that that there is no largest prime number, and Arrow's impossibility theorem. There can be multiple claims within a debate. Nevertheless, whoever makes a claim carries the burden of proof regardless of positive or negative content in the claim.
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The curve used is y^2 = x^3 + 486662x^2 + x, a Montgomery curve, over the prime field defined by the prime number 2^{255} - 19, and it uses the base point x = 9. This point generates a cyclic subgroup whose order is the prime 2^{252} + 27742317777372353535851937790883648493 and is of index 8. Using a prime order subgroup prevents mounting a Pohlig–Hellman algorithm attack. The protocol uses compressed elliptic point (only X coordinates), so it allows efficient use of the Montgomery ladder for ECDH, using only XZ coordinates.
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Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms Q in four variables which do not represent zero. One family of examples is given by :Q(x1,x2,x3,x4) = x12 \+ x22 − p(x32 \+ x42), where p is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that if the sum of two perfect squares is divisible by such a p then each summand is divisible by p.
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The absolute value of the Gamma function on the complex plane. One of the more colorful figures in 20th- century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory. Paul Erdős published more papers than any other mathematician in history, working with hundreds of collaborators.
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Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non- abelian group D4 of order 8 = 23 above shows.. See also for similar results. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.
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A perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly (with remainder zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect. A Mersenne prime is a prime number of the form ; for a number of this form to be prime, itself must also be prime. The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form , where is a Mersenne prime..
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In order for this generator to be secure, the prime number p needs to be large enough so that computing discrete logarithms modulo p is infeasible. To be more precise, any method that predicts the numbers generated will lead to an algorithm that solves the discrete logarithm problem for that prime. There is a paper discussing possible examples of the quantum permanent compromise attack to the Blum–Micali construction. This attacks illustrate how a previous attack to the Blum–Micali generator can be extended to the whole Blum–Micali construction, including the Blum Blum Shub and Kaliski generators.
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In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.
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A counterexample to the statement "all prime numbers are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "All natural numbers are either prime or composite" has the number 1 as a counterexample, as 1 is neither prime nor composite.
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Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y1/2/log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x. The main term on the left is Φ(1); which turns out to be the dominant terms of the prime number theorem, and the main correction is the sum over non-trivial zeros of the zeta function.
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In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
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Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. However, if Ln denotes the n-th lucky number, and pn the n-th prime, then Ln > pn for all sufficiently large n. Because of these apparent connections with the prime numbers, some mathematicians have suggested that these properties may be found in a larger class of sets of numbers generated by sieves of a certain unknown form, although there is little theoretical basis for this conjecture.
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By using a theorem by Carl Ludwig Siegel providing an upper bound for the real zeros (see Siegel zero) of Dirichlet L-functions formed with real non-principal characters, Walfisz obtained the Siegel-Walfisz theorem, from which the prime number theorem for arithmetic progressions can be deduced. By using estimates on exponential sums due to I. M. Vinogradov and , Walfisz obtained the currently best O-estimates for the remainder terms of the summatory functions of both the sum-of-divisors function \sigma and the Euler function \phi (in: "Weylsche Exponentialsummen in der neueren Zahlentheorie", see below).
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Dirichlet's theorem on primes in arithmetic progressions shows that there are an infinity of primes in each co-prime residue class, and the prime number theorem for arithmetic progressions shows that the primes are asymptotically equidistributed among the residue classes. The Bombieri–Vinogradov theorem gives a more precise measure of how evenly they are distributed. There is also much interest in the size of the smallest prime in an arithmetic progression; Linnik's theorem gives an estimate. The twin prime conjecture, namely that there are an infinity of primes p such that p+2 is also prime, is the subject of active research.
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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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Paul Pritchard, Explaining the wheel sieve, Acta Informatica 17 (1982), 477–485. Paul Pritchard, Fast compact prime number sieves (among others), Journal of Algorithms 4 (1983), 332–344. Pritchard observed that for the wheel sieves, one can reduce memory consumption while preserving Big O time complexity, but this generally comes at a cost in an increased constant factor for time per operation due to the extra complexity. Therefore, this special version is likely more of value as an intellectual exercise than a practical prime sieve with reduced real time expended for a given large practical sieving range.
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Consider the polynomial . In order for Eisenstein's criterion to apply for a prime number it must divide both non- leading coefficients and , which means only could work, and indeed it does since does not divide the leading coefficient , and its square does not divide the constant coefficient . One may therefore conclude that is irreducible over (and since it is primitive, over as well). Note that since is of degree 4, this conclusion could not have been established by only checking that has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.
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A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers. For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18.
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Rényi proved, using the large sieve, that there is a number K such that every even number is the sum of a prime number and a number that can be written as the product of at most K primes. Chen's theorem, a strengthening of this result, shows that the theorem is true for K = 2, for all sufficiently large even numbers. The case K = 1 is the still-unproven Goldbach conjecture. In information theory, he introduced the spectrum of Rényi entropies of order α, giving an important generalisation of the Shannon entropy and the Kullback–Leibler divergence.
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The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original proof was much simplified by P. M. Ross in 1975. Chen's theorem is a giant step towards the Goldbach's conjecture, and a remarkable result of the sieve methods. Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had showed there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most K primes.
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Closed geodesics are geodesics which are also closed curves—that is, they are curves that close up into loops. A closed geodesic is simple if it does not cross itself. A previous result, known as the "prime number theorem for geodesics", established that the number of closed geodesics of length less than L grows exponentially with L – it is asymptotic to e^L/L. However, the analogous counting problem for simple closed geodesics remained open, despite being "the key object to unlocking the structure and geometry of the whole surface," according to University of Chicago topologist Benson Farb.
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There are other similarities to number theory -- error estimates are improved upon, in much the same way that error estimates of the prime number theorem are improved upon. Also, there is a Selberg zeta function which is formally similar to the usual Riemann zeta function and shares many of its properties. Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that prime ideals in the ring of integers of a number field can be split (factored) in a Galois extension. See Covering map and Splitting of prime ideals in Galois extensions for more details.
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He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely. He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics.
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An example of parameters generated in this way are the prime numbers for the Internet Key Exchange (RFC 2409) which embed the digits of the mathematical constant pi in the digital representation of the prime number. Their first method prevents amortization of attack costs across many key exchanges at the risk of leaving open the possibility of a hidden attack like that described by Dan Bernstein against the NIST elliptic curves. The NUMS approach is open to amortization but generally avoids the Bernstein attack if only common mathematical constants such as pi and e are used.
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The space Qp of p-adic numbers is complete for any prime number . This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric. If is an arbitrary set, then the set of all sequences in becomes a complete metric space if we define the distance between the sequences and to be , where is the smallest index for which is distinct from , or if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space .
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In the theory of higher reciprocity laws, Alderson published necessary and sufficient conditions for 2 and 3 to be seventh powers, in modular arithmetic modulo a given prime number p. According to , "plain quasigroups were first studied by Helen Popova- Alderson, in a series of papers dating back to the early fifties". Smith cites in particular a posthumous paper and its references. In this context, a quasigroup is a mathematical structure consisting of a set of elements and a binary operation that does not necessarily obey the associative law, but where (like a group) this operation can be inverted.
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The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.
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The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy).
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Seventeen or Bust old client The goal of the project was to prove that 78557 is the smallest Sierpinski number, that is, the least odd k such that k·2n+1 is composite (i.e. not prime) for all n > 0\. When the project began, there were only seventeen values of k < 78557 for which the corresponding sequence was not known to contain a prime. For each of those seventeen values of k, the project searched for a prime number in the sequence : k·21+1, k·22+1, …, k·2n+1, … testing candidate values n using Proth's theorem.
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For a prime number p, the following are equivalent: # The modular curve X0+(p) = X0(p) / wp, where wp is the Fricke involution of X0(p), has genus zero. # Every supersingular elliptic curve in characteristic p can be defined over the prime subfield Fp. # The order of the Monster group is divisible by p. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then conjectural) existence of a sporadic simple group having exactly these primes as prime divisors.
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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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In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.
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The original technique for constructing -independent hash functions, given by Carter and Wegman, was to select a large prime number , choose random numbers modulo , and use these numbers as the coefficients of a polynomial of degree whose values modulo are used as the value of the hash function. All polynomials of the given degree modulo are equally likely, and any polynomial is uniquely determined by any -tuple of argument-value pairs with distinct arguments, from which it follows that any -tuple of distinct arguments is equally likely to be mapped to any -tuple of hash values.
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The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O( log p). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis.
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For large values of , the th telephone number is divisible by a large power of two, . More precisely, the 2-adic order (the number of factors of two in the prime factorization) of and of is ; for it is , and for it is .. For any prime number , one can test whether there exists a telephone number divisible by by computing the recurrence for the sequence of telephone numbers, modulo , until either reaching zero or detecting a cycle. The primes that divide at least one telephone number are :2, 5, 13, 19, 23, 29, 31, 43, 53, 59, ...
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Let k be a field of characteristic p, with p a prime number. A polynomial P(x) with coefficients in k is called an additive polynomial, or a Frobenius polynomial, if :P(a+b)=P(a)+P(b)\, as polynomials in a and b. It is equivalent to assume that this equality holds for all a and b in some infinite field containing k, such as its algebraic closure. Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that P(a + b) = P(a) + P(b) for all a and b in the field.
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In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1.
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In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two squares, such that for some integers , . :An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no term , where prime p \equiv 3 \pmod 4 and k is odd. This theorem supplements Fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers.
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The value of a is usually a prime number at least large enough to hold the number of different characters in the character set of potential keys. Radix conversion hashing of strings minimizes the number of collisions.Performance in Practice of String Hashing Functions Available data sizes may restrict the maximum length of string that can be hashed with this method. For example, a 128-bit double long word will hash only a 26 character alphabetic string (ignoring case) with a radix of 29; a printable ASCII string is limited to 9 characters using radix 97 and a 64-bit long word.
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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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If f(x) is a univariate polynomial over the integers, assumed to be content-free and square-free, one starts by computing a bound B such that any factor g(x) has coefficients of absolute value bounded by B. This way, if m is an integer larger than 2B, and if g(x) is known modulo m, then g(x) can be reconstructed from its image mod m. The Zassenhaus algorithm proceeds as follows. First, choose a prime number p such that the image of f(x) mod p remains square-free, and of the same degree as f(x). Then factor f(x) mod p.
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In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.
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The use of Gale's lemma for the coloring problem is due to . The moment curve has also been used in graph drawing, to show that all n-vertex graphs may be drawn with their vertices in a three-dimensional integer grid of side length O(n) and with no two edges crossing. The main idea is to choose a prime number p larger than n and to place vertex i of the graph at coordinates :(i, i 2 mod p, i 3 mod p).. Then a plane can only cross the curve at three positions. Since two crossing edges must have four vertices in the same plane, this can never happen.
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In addition to the multiplication of two elements of F, it is possible to define the product of an arbitrary element of by a positive integer to be the -fold sum : (which is an element of .) If there is no positive integer such that :, then is said to have characteristic 0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there is a positive integer satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by and the field is said to have characteristic then.
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Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements are essentially the statement and proof of the fundamental theorem. (In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic. (In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by infinite descent. Proposition 32 is derived from proposition 31, and proves that the decomposition is possible.
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In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field: Let p be a prime number and for any non-zero integer a, let , where pn is the highest power of p dividing a. In addition set For any rational number a/b, we set Then defines a metric on Q. The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
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In the form k = n, for integer n, the shape will appear similar to a flower. If n is odd, half of these will overlap, forming a flower with n petals. However, if n is even, the petals will not overlap, forming a flower with 2n petals. When d is a prime number, then n/d is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is + 1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.
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Volume 1 on elementary and additive number theory includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results.
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In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
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In a finite field, the product of two non-squares is a square; this implies that the polynomial x^4 + 1, which is irreducible over the integers, is reducible modulo every prime number. For example, :x^4 + 1 \equiv (x+1)^4 \pmod 2; :x^4 + 1 \equiv (x^2+x-1)(x^2-x-1) \pmod 3,\qquadsince 1^2 \equiv -2 \pmod 3; :x^4 + 1 \equiv (x^2+2)(x^2-2) \pmod 5,\qquadsince 2^2 \equiv -1 \pmod 5; :x^4 + 1 \equiv (x^2+3x+1)(x^2-3x+1) \pmod 7,\qquadsince 3^2 \equiv 2 \pmod 7.
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Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions. Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor function d(n) and Gauss's circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving the error estimates. The distribution of primes numbers among residue classes modulo an integer is an area of active research.
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Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular -gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if is a prime number, then a regular -gon can be constructed if and only if is a Fermat prime; in other words if \varphi(p)=p-1=2^k is a power of 2. For and primitive roots of unity admit a simple expression via square root of three, namely: : , Hence, both corresponding cyclotomic fields are identical to the quadratic field Q().
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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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The Prüfer -group with presentation , illustrated as a subgroup of the unit circle in the complex plane In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.
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Specifically, if k is a field and X is an indeterminate, then the ring of formal power series k is a regular local ring having (Krull) dimension 1. # If p is an ordinary prime number, the ring of p-adic integers is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field. # More generally, if k is a field and X1, X2, ..., Xd are indeterminates, then the ring of formal power series k is a regular local ring having (Krull) dimension d. # If A is a regular local ring, then it follows that the formal power series ring A is regular local.
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In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non- zero element of can be written as for some integer . If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of .
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However, although this argument proves that the next player can win, it does not identify a winning strategy for the player. After playing a prime number that is 5 or larger as a first move, the first player in a game of sylver coinage can always win by following this (non- constructive) ender strategy on their next turn. If there are any other winning openings, they must be 3-smooth numbers (numbers of the form for non- negative integers and ). For, if any number that is not of this form and is not prime is played, then the second player can win by choosing a large prime factor of .
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Derbyshire's book Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics was first published in hardcover in 2003 and then paperback in 2004. It focuses on the Riemann hypothesis, one of the Millennium Problems. The book is aimed, as Derbyshire puts it in his prologue, "at the intelligent and curious but nonmathematical reader ..." Prime Obsession explores such topics as complex numbers, field theory, the prime number theorem, the zeta function, the harmonic series, and others. The biographical sections give relevant information about the lives of mathematicians who worked in these areas, including Euler, Gauss, Lejeune Dirichlet, Lobachevsky, Chebyshev, Vallée- Poussin, Hadamard, as well as Riemann himself.
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The halting problem is one of the most famous problems in computer science, because it has profound implications on the theory of computability and on how we use computers in everyday practice. The problem can be phrased: : Given a description of a Turing machine and its initial input, determine whether the program, when executed on this input, ever halts (completes). The alternative is that it runs forever without halting. Here we are asking not a simple question about a prime number or a palindrome, but we are instead turning the tables and asking a Turing machine to answer a question about another Turing machine.
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One property he discovered was that the denominators of the fractions of Bernoulli numbers are always divisible by six. He also devised a method of calculating based on previous Bernoulli numbers. One of these methods follows: It will be observed that if n is even but not equal to zero, # is a fraction and the numerator of in its lowest terms is a prime number, # the denominator of contains each of the factors 2 and 3 once and only once, # is an integer and consequently is an odd integer. In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.
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In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.
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Extending Riemann's ideas, two proofs of the asymptotic law of the distribution of prime numbers were found independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function is nonzero for all complex values of the variable that have the form with . During the 20th century, the theorem of Hadamard and de la Vallée Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949).
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1911 he accidentally found the book of prime number tables written by Lehmer, a mathematician from the United States in the house of professor Gino Loria, a friend of his family, when he visited Genoa. Since then he spent many years to extend the first table in order to simplify "Eratosthenes Crivello" (sieve of Eratosthenes), a method from ancient Greece to find prime numbers. He gave his method a new name: "Neocribrum" (Novum Eratosthenes Cribrum) and he got recognition from the scientific community. Apart from that, he was, together with André Gerardin, member of a study commission of the Association française pour l'avancement des sciences (1946).
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In number theory, Dirichlet's theorem, also called the Dirichlet prime number 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. The numbers of the form a + nd form an arithmetic progression :a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers.
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In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0\. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem.
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Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials P with integer coefficients that always take on even number values are just those such that P/2 is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients. In questions of prime number theory, such as Schinzel's hypothesis H and the Bateman–Horn conjecture, it is a matter of basic importance to understand the case when P has no fixed prime divisor (this has been called Bunyakovsky's property, after Viktor Bunyakovsky).
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For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring over the real numbers by factoring out the ideal of multiples of the polynomial . Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring (see modular arithmetic). If is commutative, then one can associate with every polynomial in a polynomial function with domain and range equal to . (More generally, one can take domain and range to be any same unital associative algebra over .) One obtains the value by substitution of the value for the symbol in .
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In number theory, a prime number is called weakly prime if it becomes not prime when any one of its digits is changed to every single other digit. Decimal digits are usually assumed. The first weakly prime numbers are: :294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, ... For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b number with n digits must produce (b−1) × n composite numbers when a digit is changed. In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17−17)/99 + 21686652.
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In 2006, the journalist Kirill Ivanov met with the duo EU and showed them his musical material (at the time labeled as post-hip-hop, anti-hip-hop, deconstructivist hip-hop), which he had worked on in his free time. Ivanov first performed his material as part of 2H Company, a side-project of EU. The first performance in Moscow took place at the Avant Festival in 2006. In August, Ivanov performed at Nashestvie, after which the decision was made to record the new project's debut album. Initially, the group was known as 232,582,657 – 1, a representation of the largest known prime number at the time.
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In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) as opposed to arithmetic in a field with an infinite number of elements, like the field of rational numbers. While no finite field is infinite, there are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field.
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In practice, if some "naturally occurring" set of primes has a Dirichlet density, then it also has a natural density, but it is possible to find artificial counterexamples: for example, the set of primes whose first decimal digit is 1 has no natural density, but has Dirichlet density log(2)/log(10).This is attributed by J.-P. Serre to a private communication from Bombieri in A course in arithmetic; an elementary proof based on the prime number theorem is given in: A. Fuchs, G. Letta, Le problème du premier chiffre décimal pour les nombres premiers [The first digit problem for primes] (French) The Foata Festschrift. Electron. J. Combin.
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In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.
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Let N be a rational prime, and define :J0(N) = J as the Jacobian variety of the modular curve :X0(N) = X. There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J. There is also a Fricke involution w (and Atkin–Lehner involutions if N is composite). The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements : Tl − l - 1 for all l not dividing N, and by :w + 1.
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Given two randomly chosen integers a and b, it is reasonable to ask how likely it is that a and b are coprime. In this determination, it is convenient to use the characterization that a and b are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic). Informally, the probability that any number is divisible by a prime (or in fact any integer) p is 1/p; for example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by p is 1/p^2, and the probability that at least one of them is not is 1-1/p^2.
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In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. This definition can be regarded as a simultaneous generalization of both integral domains and simple rings. Although this article discusses the above definition, prime ring may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, for a characteristic p field (with p a prime number) the prime ring is the finite field of order p (cf.
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The even numbers form an ideal in the ring of integers,. but the odd numbers do not -- this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2.. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers.
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If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist. Example: Let p be a prime number, let k be imperfect of characteristic p, and choose a in k that is not a pth power. Let V = k[X]/((X^p-a)^2), let x = \overline Xand let T be the k-linear operator given by multiplication by x in V. This has as its invariant k-linear subspaces precisely the ideals of V viewed as a ring, which correspond to the ideals of k[X] containing (X^p-a)^2. Since X^p-a is irreducible in k[X], ideals of V are 0, S and J=(x^p-a)V.
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The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
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When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 3^2 denotes the square or second power of 3. The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic. This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ.
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The maze is beset by frequent tremors, and Leaven notices numbers inscribed in the narrow passageways between rooms. Rennes enters a room that he assumes to be safe and is killed when he is sprayed with acid, indicating that each trap uses different sensors to trigger them. Quentin believes each person was chosen to be there: He is a divorced police officer, Leaven is a young mathematics student, and Holloway is a free clinic doctor, while the surly Worth says he is only an office worker. Leaven hypothesizes that any room marked with a prime number is a trap, and they find a mentally challenged man named Kazan, whom Holloway insists they bring along.
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The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions. An important case, widely used in cryptography and coding theory, is that of finite fields of non-prime order. In fact, if is a prime number, and , the field of order is a simple algebraic extension of the prime field of elements, generated by a root of an irreducible polynomial of degree . A simple algebraic extension of a field , generated by the root of an irreducible polynomial of degree may be identified to the quotient ring K[X]/\langle p\rangle,, and its elements are in bijective correspondence with the polynomials of degree less than .
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This result is known as Euclid's lemma. Specifically, if a prime number divides L, then it must divide at least one factor of L. Conversely, if a number w is coprime to each of a series of numbers a1, a2, ..., an, then w is also coprime to their product, a1 × a2 × ... × an. Euclid's lemma suffices to prove that every number has a unique factorization into prime numbers. To see this, assume the contrary, that there are two independent factorizations of L into m and n prime factors, respectively : Since each prime p divides L by assumption, it must also divide one of the q factors; since each q is prime as well, it must be that p = q.
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The Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pn. Conversely, if a subgroup has order pn, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn. A very important consequence of Theorem 2 is that the condition np = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup (there are groups that have normal subgroups but no normal Sylow subgroups, such as S4).
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When n = 2, it is easy to see why this is incorrect: (x + y)2 can be correctly computed as x2 + 2xy + y2 using distributivity (commonly known as the FOIL method). For larger positive integer values of n, the correct result is given by the binomial theorem. The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x + y)p = xp + yp. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since p divides all the binomial coefficients apart from the first and the last, making all intermediate terms equal to zero.
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251, quoted in Ribenboim defines a triply palindromic prime as a prime p for which: p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime.Paulo Ribenboim, The New Book of Prime Number Records For example, p = 1011310 \+ 4661664 + 1, which has q = 11311 digits, and 11311 has r = 5 digits. The first (base-10) triply palindromic prime is the 11-digit 10000500001. It's possible that a triply palindromic prime in base 10 may also be palindromic in another base, such as base 2, but it would be highly remarkable if it were also a triply palindromic prime in that base as well.
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Worry beads may be constructed from any type of bead, although amber, amber resin (such as faturan), and coral are preferred, as they are thought to be more pleasant to handle than non-organic materials such as metal or minerals. Greek worry beads generally have an odd number of beads, often one more than a multiple of four (e.g. (4×4)+1, (5×4)+1, and so on) or a prime number (usually 17, 19 or 23), and usually have a head composed of a fixed bead (παπάς "priest"), a shield (θυρεός) to separate the two threads and help the beads to flow freely, and a tassel (φούντα). Usually the length of worry beads is approximately two palm widths.
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Additionally, he writes of a few other mathematical concepts, including Null hypothesis and the Quantic Formula. CHAPTER 8, REDUCTIO AD UNLIKELY: This chapter focuses on the works and theorems/concepts of many famous mathematicians and philosophers. These include but aren’t limited to the Reductio Ad Absurdum by Aristotle, a look into the constellation Taurus by John Mitchell, and Yitang “Tom” Zhangs “bounded gaps” conjecture. He also delves into explaining rational numbers, the prime number theorem, and makes up his own word, “flogarithms”. CHAPTER 9, THE INTERNATIONAL JOURNAL OF HARUSPICY: Ellenberg relates the practice of Haruspicy, genes that affect Schizophrenia, and the accuracy of published papers as well as other things to the “P value” or Statistical Significance.
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In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. introduced Artin–Schreier theory for extensions of prime degree p, and generalized it to extensions of prime power degree pn. If K is a field of characteristic p, a prime number, any polynomial of the form :X^p - X + \alpha,\, for \alpha in K, is called an Artin–Schreier polynomial. When \alpha eq \beta^p-\beta for all \beta \in K, this polynomial is irreducible in K[X], and its splitting field over K is a cyclic extension of K of degree p.
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In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power of , which can be found by iteratively "lifting" the solution modulo successive powers of . More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of Newton's method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial.
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It can be proven mathematically that all pairs of valid ISBN-10s differ in at least two digits. It can also be proven that there are no pairs of valid ISBN-10s with eight identical digits and two transposed digits. (These proofs are true because the ISBN is less than eleven digits long and because 11 is a prime number.) The ISBN check digit method therefore ensures that it will always be possible to detect these two most common types of error, i.e., if either of these types of error has occurred, the result will never be a valid ISBN – the sum of the digits multiplied by their weights will never be a multiple of 11.
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An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite. The first few isolated primes are :2, 23, 37, 47, 53, 67, 79, 83, 89, 97, ... It follows from Brun's theorem that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold n and the number of all primes less than n tends to 1 as n tends to infinity.
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The conjecture states that, for every integer x > 1, there is at least one prime number between : x(x − 1) and x2, and at least another prime between : x2 and x(x + 1). It can also be phrased equivalently as stating that the prime-counting function must take unequal values at the endpoints of each range.. That is: : π(x2 − x) < π(x2) < π(x2 + x) for x > 1 with π(x) being the number of prime numbers less than or equal to x. The end points of these two ranges are a square between two pronic numbers, with each of the pronic numbers being twice a pair triangular number. The sum of the pair of triangular numbers is the square.
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For example consider , in which the coefficient 1 of is not divisible by any prime, Eisenstein's criterion does not apply to . But if one substitutes for in , one obtains the polynomial , which satisfies Eisenstein's criterion for the prime number . Since the substitution is an automorphism of the ring , the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally. In this particular example it would have been simpler to argue that (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein's criterion apply is a useful way to broaden its scope.
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This includes his paper on what is now known as the Piatetski-Shapiro prime number theorem, which states that, for 1 ≤ c ≤ 12/11, the number of integers 1 ≤ n ≤ x for which the integer part of nc is prime is asymptotically x / c log x as x → ∞. After leaving the Moscow Pedagogical Institute, he spent a year at the Steklov Institute, where he received the advanced Doctor of Sciences degree, also in 1954, under the direction of Igor Shafarevich. His contact with Shafarevich, who was a professor at the Steklov Institute, broadened Ilya's mathematical outlook and directed his attention to modern number theory and algebraic geometry. This led, after a while, to the influential joint paper in which they proved a Torelli theorem for K3 surfaces.
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A subgroup, , of an abelian group, , is called p-basic, for a fixed prime number, , if the following conditions hold: # is a direct sum of cyclic groups of order and infinite cyclic groups; # is a p-pure subgroup of ; # The quotient group, , is a p-divisible group. Conditions 1–3 imply that the subgroup, , is Hausdorff in the p-adic topology of , which moreover coincides with the topology induced from , and that is dense in . Picking a generator in each cyclic direct summand of creates a p-basis of B, which is analogous to a basis of a vector space or a free abelian group. Every abelian group, , contains p-basic subgroups for each , and any 2 p-basic subgroups of are isomorphic.
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For non-separable extensions, necessarily in characteristic p with p a prime number, then at least when the degree [L : K] is p, L / K has a primitive element, because there are no intermediate subfields. When [L : K] = p2, there may not be a primitive element (and therefore there are infinitely many intermediate fields). This happens, for example if K is :Fp(T, U), the field of rational functions in two indeterminates T and U over the finite field with p elements, and L is obtained from K by adjoining a p-th root of T, and of U. In fact one can see that for any α in L, the element αp lies in K, but a primitive element must have degree p2 over K.
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A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number. The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation r2 + s4 = t4 and proving Fermat's theorem on sums of two squares, which states that an odd prime p can be expressed as a sum of two squares when p ≡ 1 (mod 4) (see proof). In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression).
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This is the explicit form in this case of the abstract result that over an algebraically closed field K (such as the complex numbers) the regular representation of G is completely reducible, provided that the characteristic of K (if it is a prime number p) doesn't divide the order of G. That is called Maschke's theorem. In this case the condition on the characteristic is implied by the existence of a primitive n-th root of unity, which cannot happen in the case of prime characteristic p dividing n. Circulant determinants were first encountered in nineteenth century mathematics, and the consequence of their diagonalisation drawn. Namely, the determinant of a circulant is the product of the n eigenvalues for the n eigenvectors described above.
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The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a 512-bit prime number, so called export grade. The authors needed several thousand CPU cores for a week to precompute data for a single 512-bit prime. Once that was done, individual logarithms could be solved in about a minute using two 18-core Intel Xeon CPUs. As estimated by the authors behind the Logjam attack, the much more difficult precomputation needed to solve the discrete log problem for a 1024-bit prime would cost on the order of $100 million, well within the budget of a large national intelligence agency such as the U.S. National Security Agency (NSA).
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For proving bounds on this problem, it may be assumed without loss of generality that the inputs are strings over a two-letter alphabet. For, if two strings over a larger alphabet differ then there exists a string homomorphism that maps them to binary strings of the same length that also differ. Any automaton that distinguishes the binary strings can be translated into an automaton that distinguishes the original strings, without any increase in the number of states.. It may also be assumed that the two strings have equal length. For strings of unequal length, there always exists a prime number whose value is logarithmic in the smaller of the two input lengths, such that the two lengths are different modulo .
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In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group. Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold. The naming of the conjecture is for David Hilbert, and the American topologist Paul A. Smith.
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In the last section of the DisquisitionesGauss, DA. The 7th § is arts. 336–366Gauss proved if satisfies certain conditions then the -gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the -gon is constructible, then must satisfy Gauss's conditions Gauss provesGauss, DA, art 366 that a regular -gon can be constructed with straightedge and compass if is a power of 2. If is a power of an odd prime number the formula for the totient says its totient can be a power of two only if is a first power and is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537.
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He proved the Waring's problem for k\ge 6 in 1935 under the further condition of (3^k + 1)/(2^k - 1)\le [1.5^k] + 1 ahead of Leonard Eugene Dickson who around the same time proved it for k\ge 7. He showed that g(k) = 2^k + l - 2 where l is the largest natural number \le (3/2)^k and hence computed the precise value of g(6) = 73. The Pillai sequence 1, 4, 27, 1354, ..., is a quickly-growing integer sequence in which each term is the sum of the previous term and a prime number whose following prime gap is larger than the previous term. It was studied by Pillai in connection with representing numbers as sums of prime numbers.
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A palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: :2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. The largest known is (474,501 digits): :10474500 \+ 999 × 10237249 \+ 1.
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In the infinite case, the product is interpreted in the sense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K are finite. If M/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between M and K, via simple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediate field L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and [L:K] = p, in which case L is equal to M. Therefore, there are no intermediate fields (apart from M and K themselves).
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In January 1999, while preparing some work for his students, he identified a highly structured prime number with exactly two thousand digits. Dubbing this prime a millennium prime, he wrote an email about it to a niece and nephew, which was subsequently published by Folding Landscapes, the publishing house of the cartographer Tim Robinson. He donated his author royalties to the Irish Cancer Society, and subsequently wrote an Irishman's Diary column about it for the Irish Times newspaper. In July 1999 – while a participant in the Proth Search Group – he became the discoverer of the then-largest known composite Fermat number, a record which his St. Patrick's College (Drumcondra) based Proth- Gallot Group twice broke in 2003, the 1999 record having stood until then.
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Paul Trevier Bateman (June 6, 1919 – December 26, 2012Paul Bateman (obituary), News-Gazette, 12-27-2012.) was an American number theorist, known for formulating the Bateman–Horn conjecture on the density of prime number values generated by systems of polynomials and the New Mersenne conjecture relating the occurrences of Mersenne primes and Wagstaff primes. Born in Philadelphia, Bateman received his Ph.D. from the University of Pennsylvania in 1946, under the supervision of Hans Rademacher. After temporary positions at Yale University and the Institute for Advanced Study, he joined in 1950 the mathematics department at the University of Illinois at Urbana-Champaign, where he was department chair for 15 years and was subsequently an emeritus professor. He was the doctoral advisor of 20 students, including Marvin Knopp, Kevin McCurley, and George B. Purdy.
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This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p is the smallest prime number dividing the order of a finite group, G, then if has order p, H must be a normal subgroup of G. Given G and a normal subgroup N, then G is a group extension of by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and . This is a special case of the extension problem. An example where the extension is not split is as follows: Let G = Z4 = {0, 1, 2, 3}, and N = {0, 2}, which is isomorphic to Z2.
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While each quarter would be equal in length (13 weeks), thirteen is a prime number, placing all activities currently done on a quarterly basis out of alignment with the months. Christian, Islamic and Jewish leaders are historically opposed to the calendar, as their tradition of worshiping every seventh day would result in either the day of the week of worship changing from year to year, or eight days passing when "The Festival of All the Dead” or “The Festival of Holy Women" occurs. Birthdays, significant anniversaries, and other holidays would need to be recalculated as a result of a calendar reform, and would always be on the same day of the week. This could be problematic for Public holidays that would fall under non- working days under the new system; eg.
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The most commonly used case of Cohen's theorem is when the complete Noetherian local ring contains some field. In this case Cohen's structure theorem states that the ring is of the form k'x1,...,xn/(I) for some ideal I, where k is its residue class field. In the unequal characteristic case when the complete Noetherian local ring does not contain a field, Cohen's structure theorem states that the local ring is a quotient of a formal power series ring in a finite number of variables over a Cohen ring with the same residue field as the local ring. A Cohen ring is a field or a complete characteristic zero discrete valuation ring whose maximal ideal is generated by a prime number p (equal to the characteristic of the residue field).
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In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbers Qp or its algebraic closure. The source of a p-adic L-function tends to be one of two types. The first source--from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a p-adic L-function --is via the p-adic interpolation of special values of L-functions.
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For, if not, it would be congruent to 1 mod 3 and 2p + 1 would be congruent to 3 mod 3, impossible for a prime number.. Similar restrictions hold for larger prime moduli, and are the basis for the choice of the "correction factor" 2C in the Hardy–Littlewood estimate on the density of the Sophie Germain primes. If a Sophie Germain prime p is congruent to 3 (mod 4) (, Lucasian primes), then its matching safe prime 2p + 1 will be a divisor of the Mersenne number 2p − 1\. Historically, this result of Leonhard Euler was the first known criterion for a Mersenne number with a prime index to be composite.. It can be used to generate the largest Mersenne numbers (with prime indices) that are known to be composite..
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A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10). 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 () Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 (OEIS really calls this sequence circular primes, but not the above sequence): 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 () All repunit primes are circular.
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Most early Greeks did not even consider 1 to be a number, For a selection of quotes from and about the ancient Greek positions on this issue, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6. so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number. By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number., pp. 7–13.
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The 3-adic integers, with selected corresponding characters on their Pontryagin dual group In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.
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The prime number theorem (PNT) implies that the number of primes up to x is roughly x/ln(x), so if we replace x with 2x then we see the number of primes up to 2x is asymptotically twice the number of primes up to x (the terms ln(2x) and ln(x) are asymptotically equivalent). Therefore the number of primes between n and 2n is roughly n/ln(n) when n is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's Postulate. So Bertrand's postulate is comparatively weaker than the PNT. But PNT is a deep theorem, while Bertrand's Postulate can be stated more memorably and proved more easily, and also makes precise claims about what happens for small values of n.
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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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The p-rank of an abelian variety A over a field K of characteristic p is the integer k for which the kernel A[p] of multiplication by p has pk points. It may take any value from 0 to d, the dimension of A; by contrast for any other prime number l there are l2d points in A[l]. The reason that the p-rank is lower is that multiplication by p on A is an inseparable isogeny: the differential is p which is 0 in K. By looking at the kernel as a group scheme one can get the more complete structure (reference David Mumford Abelian Varieties pp. 146–7); but if for example one looks at reduction mod p of a division equation, the number of solutions must drop.
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Super-prime numbers (also known as higher-order primes or prime-indexed primes or PIPs) are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins :3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... . That is, if p(i) denotes the ith prime number, the numbers in this sequence are those of the form p(p(i)). used a computer-aided proof (based on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers.
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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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This program can be used to strip Cascading Style Sheets tags from HTML pages. In one case, a school removed a student's webpage that included a copy of this program, mistaking it for the original DeCSS program, and received a great deal of negative media attention. The CSS stripping program had been specifically created to bait the MPAA in this manner. In protest against legislation that prohibits publication of copy protection circumvention code in countries that implement the WIPO Copyright Treaty (such as the United States' Digital Millennium Copyright Act), some have devised clever ways of distributing descriptions of the DeCSS algorithm, such as through steganography, through various Internet protocols, on T-shirts and in dramatic readings, as MIDI files, as a haiku poem (DeCSS haiku), and even as a so-called illegal prime number.
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Cheryl first says that she is the oldest of three siblings, and that their ages multiplied makes 144. 144 can be decomposed into prime number factors by the fundamental theorem of arithmetic (), and all possible ages for Cheryl and her two brothers examined (for example, 16, 9, 1, or 8, 6, 3, and so on). The sums of the ages can then be computed. Because Bernard (who knows the bus number) cannot determine Cheryl's age despite having been told this sum, it must be a sum that is not unique among the possible solutions. On examining all the possible ages, it turns out there are two pairs of sets of possible ages that produce the same sum as each other: 9, 4, 4 and 8, 6, 3, which sum to 17, and 12, 4, 3 and 9, 8, 2, which sum to 19.
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In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (the product of the first n primes). Primality tests show that : pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... : pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ... The first term of the second sequence is 0, because p0# = 1 is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, as p1# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are :2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 , the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, found by the PrimeGrid project.Primegrid.
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120 is the factorial of 5 and one less than a square, making (5, 11) a Brown number pair. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8 + 16 + 32 + 64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number, being adjacent to 119 = 7 × 17 and 121 = 112.
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In number theory, a Wieferich prime is a prime number p such that p2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.
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If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1.
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Paul Erdős (in ) observed that, when n is a prime number, the set of n grid points (i, i2 mod n), for 0 ≤ i < n, contains no three collinear points. When n is not prime, one can perform this construction for a p × p grid contained in the n × n grid, where p is the largest prime that is at most n. As a consequence, for any ε and any sufficiently large n, one can place :(1 - \epsilon)n points in the n × n grid with no three points collinear. Erdős' bound has been improved subsequently: show that, when n/2 is prime, one can obtain a solution with 3(n - 2)/2 points by placing points on the hyperbola xy ≡ k (mod n/2), where k may be chosen arbitrarily as long as it is nonzero mod n/2.
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Pierre de Fermat Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following: > If is a prime and is any integer not divisible by , then is divisible by . Fermat's original statement was > This may be translated, with explanations and formulas added in brackets for > easier understanding, as: > Every prime number [] divides necessarily one of the powers minus one of any > [geometric] progression [] [that is, there exists such that divides ], and > the exponent of this power [] divides the given prime minus one [divides ]. > After one has found the first power [] that satisfies the question, all > those whose exponents are multiples of the exponent of the first one satisfy > similarly the question [that is, all multiples of the first have the same > property].
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The Miller–Rabin primality test uses the following extension of Fermat's little theorem: > If is an odd prime number, and , with odd, then for every prime to , either > , or there exists such that and This result may be deduced from Fermat's little theorem by the fact that, if is an odd prime, then the integers modulo form a finite field, in which has exactly two square roots, 1 and −1. The Miller–Rabin test uses this property in the following way: given , with odd, an odd integer for which primality has to be tested, choose randomly such that ; then compute ; if is not 1 nor −1, then square it repeatedly modulo until you get 1, −1, or have squared times. If and −1 has not been obtained, then is not prime. Otherwise, may be prime or not.
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The first of the four to be composed, this étude alternates two separate sets of refrains with longer strophes given over to the rhythmic neumes of the title. The first set of refrains is marked "rythme en ligne triple: 1 à 5, 6 à 10, 11 à 15", which means there are three durations, short, medium, and long, which are progressively expanded, upon each repetition, by the addition of a semiquaver unit. The second set of refrains is marked "Nombre premier en rythme non rétrogradable" (A prime number in non-retrogradable rhythm) and, like the first set of refrains with which they alternate, are expanded upon repetition—in this case by a series of progressively larger prime numbers: 41, 43, 47, and 53 semiquavers. The strophes which occur between the refrains are marked "neumes rythmiques, avec résonances et intensités fixes" (rhythmic neumes, with fixed resonances and intensities).
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In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers.
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The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but F_5 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular n-gon is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes. Likewise, a regular n-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2^a3^b+1. It is possible to partition any convex polygon into n smaller convex polygons of equal area and equal perimeter, when n is a power of a prime number, but this is not known for other values of n.
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All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form ⟨m⟩ = mZ, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility.. Thus, since a prime number p has no nontrivial divisors, pZ is a maximal proper subgroup, and the quotient group Z/pZ is simple; in fact, a cyclic group is simple if and only if its order is prime.. All quotient groups Z/nZ are finite, with the exception For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, generated by the residue class of n/d. There are no other subgroups.
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In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).
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Dickson considered himself a Texan by virtue of having grown up in Cleburne, where his father was a banker, merchant, and real estate investor. He attended the University of Texas at Austin, where George Bruce Halsted encouraged his study of mathematics. Dickson earned a B.S. in 1893 and an M.S. in 1894, under Halsted's supervision. Dickson first specialised in Halsted's own specialty, geometry.A. A. Albert (1955) Leonard Eugene Dickson 1874–1954 from National Academy of Sciences Both the University of Chicago and Harvard University welcomed Dickson as a Ph.D. student, and Dickson initially accepted Harvard's offer, but chose to attend Chicago instead. In 1896, when he was only 22 years of age, he was awarded Chicago's first doctorate in mathematics, for a dissertation titled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group, supervised by E. H. Moore.
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The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as Y^2 = X(X-1)(X-2)\ldots (X-k). Bounds for the zeroes of the local zeta-function immediately imply bounds for sums \sum \chi(X(X-1)(X-2)\ldots (X-k)), where χ is the Legendre symbol modulo a prime number p, and the sum is taken over a complete set of residues mod p. In the light of this connection it was appropriate that, with a Trinity research fellowship, Davenport in 1932-1933 spent time in Marburg and Göttingen working with Helmut Hasse, an expert on the algebraic theory. This produced the work on the Hasse–Davenport relations for Gauss sums, and contact with Hans Heilbronn, with whom Davenport would later collaborate.
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A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. A circular prime with at least two digits can only consist of combinations of the digits 1, 3, 7 or 9, because having 0, 2, 4, 6 or 8 as the last digit makes the number divisible by 2, and having 0 or 5 as the last digit makes it divisible by 5. The complete listing of the smallest representative prime from all known cycles of circular primes (The single- digit primes and repunits are the only members of their respective cycles) is 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R19, R23, R317, R1031, R49081, R86453, R109297, and R270343, where Rn is a repunit prime with n digits.
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In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p.. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater... The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric point of view, L corresponds to another curve C′, together with a morphism :π : C′ → C. Geometrically, the assertion that π is ramified at a finite set S of points on C means that π restricted to the complement of S in C is an étale morphism. This is in analogy with the case of Riemann surfaces.
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The plot revolves around three central human characters, George Griffin, Roger Coulton, and Alice Lang. Set from 1987 to 2004, the book details the efforts of physicists George and Roger as they work to bring the Superconducting Super Collider (SSC) online in Waxahachie, Texas. Alice is a novelist, working on her latest horror work, who becomes involved as she researches material for her book at the SSC. She and George fall in love just as preliminary trial runs of the SSC produce an unexplained phenomenon: a Snark, to borrow an expression from Lewis Carroll, or an impossible event, in the form of a heavy particle which emerged from the planned head-on collision between two 20 TeV protons inside the SSC. In addition to violating physical laws such as the conservation of mass, this particle emits pulses of radioactivity, spelling out the numerical prime number sequence of 2-3-5-7-11-13-17-19-23-29-31-37.
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Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The outer automorphism group is isomorphic to the semidirect product D \rtimes (F \times G) where all these groups D, F, G are cyclic of the respective orders d, f, g, except for type D_n(q), q odd, where the group of order d=4 is C_2 \times C_2, and (only when n=4) G = S_3, the symmetric group on three elements. The notation (a,b) represents the greatest common divisor of the integers a and b.
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Most commonly, the modulus is chosen as a prime number, making the choice of a coprime seed trivial (any 0 < X < m will do). This produces the best quality output, but introduces some implementation complexity and the range of the output is unlikely to match the desired application; converting to the desired range requires an additional multiplication. Using a modulus m which is a power of two makes for a particularly convenient computer implementation, but comes at a cost: the period is at most m/4, and the low bits have periods shorter than that. This is because the low k bits form a modulo-2 generator all by themselves; the higher-order bits never affect lower-order bits. The values X are always odd (bit 0 never changes), bits 2 and 1 alternate (the low 3 bits repeat with a period of 2), the low 4 bits repeat with a period of 4, and so on.
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The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13 For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime.
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Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to m X^2 + n Y^2)—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation a x^2 + b y^2 + c z^2 = 0 and worked on quadratic forms along the lines later developed fully by Gauss.
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In 1954, he was awarded the Bôcher Memorial Prize of the American Mathematical Society and in 1971 the Chauvenet Prize (after winning in 1970 the Lester R. Ford Award) of the Mathematical Association of America for his paper A Motivated Account of an Elementary Proof of the Prime Number Theorem. In 1974 he published a paper proving that more than a third of the zeros of the Riemann zeta function lie on the critical line, a result later improved to two fifths by Conrey. He received both his bachelor's degree and his master's degree in electrical engineering from MIT in 1934, where he had studied under Norbert Wiener and took almost all of the graduate-level courses in mathematics. He received the MIT Redfield Proctor Traveling Fellowship to study at the University of Cambridge, with the assurance that MIT would reward him with a Ph.D. upon his return regardless of whatever he produced at Cambridge.
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Demonstration, with Cuisenaire rods, of the powerful nature of 1, 4, 8, and 9 A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square- full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful. The following is a list of all powerful numbers between 1 and 1000: :1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... .
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Suppose the hotel is next to an ocean, and an infinite number of car ferries arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This is a situation involving three "levels" of infinity, and it can be solved by extensions of any of the previous solutions. The prime factorization method can be applied by adding a new prime number for every additional layer of infinity ( 2^s 3^c 5^f, with f the ferry). The prime power solution can be applied with further exponentiation of prime numbers, resulting in very large room numbers even given small inputs. For example, the passenger in the second seat of the third bus on the second ferry (address 2-3-2) would raise the 2nd odd prime (5) to 49, which is the result of the 3rd odd prime (7) being raised to the power of his seat number (2).
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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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Buffalo had experienced more success on the field and at the gate than Baltimore, and was also a larger market at the time (and would not have to share their territory with an established team as Baltimore would with the Washington Redskins). Additionally, the original three-team plan would have left the league with 13 teams, not only an odd number and prime number that made making equal divisions impossible, but also one considered to be bad luck. The move had left Buffalo as the only AAFC market without an NFL team post-merger, and one that had outdrawn the NFL average in fan attendance. With that in mind, Buffalo fans produced more than 15,000 season ticket pledges, raised $175,000 in a stock offering,The Coffin Corner, Volume 19, 1997, published by the Professional Football Researchers Association, The Other Buffalo Bills, by Joe Marren and filed a separate application to join. When the vote to admit Buffalo was held on January 20, 1950, a majority of league owners (including the three already-admitted AAFC teams) were willing to accept Buffalo.
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In a sociable chain, or aliquot cycle, a sequence of divisor-sums returns to the initial number. These are the two chains Poulet described in 1918: 12496 → 14288 → 15472 → 14536 → 14264 → 12496 (5 links) 14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (28 links) The second chain remains by far the longest known, despite the exhaustive computer searches begun by the French mathematician Henri Cohen in 1969. Poulet introduced sociable chains in a paper in the journal L'Intermédiaire des Mathématiciens #25 (1918). The paper ran like this: :If one considers a whole number a, the sum b of its proper divisors, the sum c of the proper divisors of b, the sum d of the proper divisors of c, and so on, one creates a sequence that, continued indefinitely, can develop in three ways: :The most frequent is to arrive at a prime number, then at unity [i.e.
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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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Five is conjectured to be the only odd untouchable number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two consecutive primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being plus . It is the only Fibonacci number that is equal to its position. 5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... ( lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers. 5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20, 5 is a 1-automorphic number.
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In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q. For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both have period 18; 3541 and 27961 both have period 20.
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The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then a residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q), so that for any integer x, one of x and −x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a2 (mod q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2, and again, r ≡ (−a)2 ≡ b4 (mod q) is a biquadratic residue.Gauss, BQ § 3 Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4). Gauss provedGauss, BQ §§ 4–7 that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing (p−1)/4 numbers.
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Specifically, Carmody applied Dirichlet's theorem to several prime candidates of the form k·256n \+ b, where k was the decimal representation of the original compressed file. Multiplying by a power of 256 adds as many trailing null characters to the gzip file as indicated in the exponent which would still result in the DeCSS C code when unzipped. Of those prime candidates, several were identified as probable prime using the open source program OpenPFGW, and one of them was proved prime using the ECPP algorithm implemented by the Titanix software.DVD descrambler encoded in ‘illegal’ prime number (Thomas C. Greene, The Register, Mon 19 March 2001) Even at the time of discovery in 2001, this 1401-digit number, of the form k·2562 \+ 2083, was too small to be mentioned, so Carmody discovered a 1905-digit prime, of the form k·256211 \+ 99, that was the tenth largest prime found using ECPP, a remarkable achievement by itself and worthy of being published on the lists of the highest prime numbers.
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For example, Z/12Z is isomorphic to the direct product Z/3Z × Z/4Z under the isomorphism (k mod 12) → (k mod 3, k mod 4); but it is not isomorphic to Z/6Z × Z/2Z, in which every element has order at most 6. If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when .. The cyclic numbers include all primes, but some are composite such as 15\. However, all cyclic numbers are odd except 2. The cyclic numbers are: :1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... The definition immediately implies that cyclic groups have group presentation and for finite n..
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After a copy of "The Library of Babel" itself, as translated into English by Andrew Hurley, The Unimaginable Mathematics of Borges' Library of Babel has seven chapters on its mathematics. The first chapter, on combinatorics, repeats the calculation above, of the number of books in the library, putting it in context with the size of the known universe and with other huge numbers, and uses this material as an excuse to branch off into a discussion of logarithms and their use in estimation. The second chapter concerns a line in the story about the existence of a library catalog for the library, using information theory to prove that such a catalog would necessarily equal in size the library itself, and touching on topics including the prime number theorem. The third chapter considers the mathematics of the infinite, and the possibility of books with infinitely many, infinitely thin pages, connecting these topics both to a footnote in "The Library of Babel" and to another Borges story, "The Book of Sand", about such an infinite book.
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Gauss with his 1801 book Disquisitiones Arithmeticae. In this book, Gauss used the fundamental theorem for proving the law of quadratic reciprocity. In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1Using the empty product rule one need not exclude the number 1, and the theorem can be stated as: every positive integer has unique prime factorization. either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors. For example, : 1200 = 2^4 \cdot 3 \cdot 5^2 = (2 \cdot 2 \cdot 2 \cdot 2) \cdot 3 \cdot (5 \cdot 5) = 5 \cdot 2 \cdot 5 \cdot 2 \cdot 3 \cdot 2 \cdot 2 = \ldots The theorem says two things for this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.
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Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divide the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable (which is the case for every p but a finite number) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P. Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of P, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then X^2-D is a resolvent for the alternating group. This resolvent is always separable (assuming the characteristic is not 2) if P is irreducible, but most resolvents are not always separable.
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In an attempt to make Turing machine more intuitive, Z. A. Melzac consider the task of computing with positive numbers. The machine has an infinite abacus, an infinite number of counters (pebbles, tally sticks) initially at a special location S. The machine is able to do one operation: > Take from location X as many counters as there are in location Y and > transfer them to location Z and proceed to next instruction. If this > operation is not possible because there is not enough counters in Y, then > leave the abacus as it is and proceed to instruction T. This essentially a subneg where the test is done before rather than after the subtraction, in order to keep all numbers positive and mimic a human operator computing on a real world abacus. Pseudocode: command X, Y, Z, T ; if (Mem[Y] < Mem[X]) goto T ; Mem[Z] = Mem[Y] - Mem[X] After giving a few programs: multiplication, gcd, computing the n-th prime number, representation in base b of an arbitrary number, sorting in order of magnitude, Melzac shows explicitly how to simulate an arbitrary Turing machine on his arithmetic machine.
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In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes: :2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 . For example, 409 is a minimal prime because there is no prime among the shorter subsequences of the digits: 4, 0, 9, 40, 49, 09. The subsequence does not have to consist of consecutive digits, so 109 is not a minimal prime (because 19 is prime). But it does have to be in the same order; so, for example, 991 is still a minimal prime even though a subset of the digits can form the shorter prime 19 by changing the order. Similarly, there are exactly 32 composite numbers which have no shorter composite subsequence: :4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731 . There are 146 primes congruent to 1 mod 4 which have no shorter prime congruent to 1 mod 4 subsequence: :5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, ... There are 113 primes congruent to 3 mod 4 which have no shorter prime congruent to 3 mod 4 subsequence: :3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, ...
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The primitive part of the Fibonacci numbers are :1, 1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 46, 15005, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 321, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, ... The product of the primitive prime factors of the Fibonacci numbers are :1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 23, 3001, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 107, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, 64079, 2971215073, 1103, 598364773, 15251, ... The first case of more than one primitive prime factor is 4181 = 37 × 113 for F_{19}. The primitive part has a non-primitive prime factor in some cases. The ratio between the two above sequences is :1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, .... The natural numbers n for which F_n has exactly one primitive prime factor are :3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 60, 62, 63, 65, 66, 72, 74, 75, 76, 82, 83, 93, 94, 98, 105, 106, 108, 111, 112, 119, 121, 122, 123, 124, 125, 131, 132, 135, 136, 137, 140, 142, 144, 145, ... If and only if a prime p is in this sequence, then F_p is a Fibonacci prime, and if and only if 2p is in this sequence, then L_p is a Lucas prime (where L_n is the Lucas sequence), and if and only if 2n is in this sequence, then L_{2^{n-1}} is a Lucas prime. Number of primitive prime factors of F_n are :0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, ... The least primitive prime factor of F_n are :1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101, ...
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