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Wilson Prime
In number theory, a Wilson prime is a prime number p such that p^2 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. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham. The only known Wilson primes are 5, 13, and 563 . Costa et al. Write that "the case p=5 is trivial", and credit the observation that 13 is a Wilson prime to . Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval ,y/math> is about \log\log_x y. Several compute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wilson (Mathematician)
John Wilson (6 August 1741, Applethwaite, Westmorland – 18 October 1793, Kendal, Westmorland) Robinson (2003), p. 50 was an English mathematician and judge. Wilson's theorem is named after him. Wilson attended school in Staveley, Cumbria before going up to Peterhouse, Cambridge in 1757, where he was a student of Edward Waring. He was Senior Wrangler in 1761. He was later knighted, and became a Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemat ... in 1782. He was Judge of Common Pleas from 1786 until his death in 1793. See also * Wilson prime Notes References * C. M. Neale (1907) ''The Senior Wranglers of the University of Cambridge''. Availablonline* Robinson, Derek John Scott. ''An introduction to abstract algebra''. 2003. Walter de Gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Zimmermann (mathematician)
Paul Zimmermann (born 13 November 1964) is a French computational mathematician, working at INRIA. Zimmermann co-authored the book Computational Mathematics with SageMath used by Mathematical students worldwide. His interests include asymptotically fast arithmetic—he wrote a book on algorithms for computer arithmetic with Richard Brent. He has developed some of the fastest available code for manipulating polynomials over GF(2), and for calculating hypergeometric constants to billions of decimal places. He is associated with the CARAMEL project to develop efficient arithmetic, in a general context and in particular in the context of algebraic curves of small genus; arithmetic on polynomials of very large degree turns out to be useful in algorithms for point-counting on such curves. He is also interested in computational number theory. In particular, he has contributed to some of the record computations in integer factorisation and discrete logarithm. He has been an acti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 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. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronger v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wall–Sun–Sun Prime
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. Definition Let p be a prime number. When each term in the sequence of Fibonacci numbers F_n is reduced modulo p, the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p). Since F_0 = 0, it follows that ''p'' divides F_. A prime ''p'' such that ''p''2 divides F_ is called a Wall–Sun–Sun prime. Equivalent definitions If \alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index m such that m divides F_), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p^2 divides F_. For a prime ''p'' ≠ 2, 5, the rank of apparition \alpha(p) is known to divide p - \left(\tfrac\right), where the Legendre symbol \textstyle\left(\frac\right) has the values :\left(\frac\right) = \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Congruences
In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences. Table of congruences characterizing special primes Other prime-related congruences There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers. Many of these alternate statements characterizing primality are related to Wilson's theorem, or are restatements of this classical result given in terms of other special variants of generalized factorial functions. For instance, new variants of Wilson's theorem stated in terms of the hyperfactorials, subfactorials, and superfactorials are given in. Variants of Wilson's theorem For integers k \geq 1, we have the following form of Wilson's theorem: :(k-1)! (p-k)! \equiv (-1)^k \pmod \iff p \text If p is odd, we have that :\left(\frac\right)!^2 \equiv (-1)^ \pmod \iff p \text Clem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to 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. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson Quotient
The Wilson quotient ''W''(''p'') is defined as: :W(p) = \frac If ''p'' is a prime number, the quotient is an integer by Wilson's theorem; moreover, if ''p'' is composite, the quotient is not an integer. If ''p'' divides ''W''(''p''), it is called a Wilson prime. The integer values of ''W''(''p'') are : : ''W''(2) = 1 : ''W''(3) = 1 : ''W''(5) = 5 : ''W''(7) = 103 : ''W''(11) = 329891 : ''W''(13) = 36846277 : ''W''(17) = 1230752346353 : ''W''(19) = 336967037143579 : ... It is known that :W(p)\equiv B_-B_\pmod, :p-1+ptW(p)\equiv pB_\pmod{p^2}, where B_k is the ''k''-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting t=1 and t=2. See also * Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref> The smallest solutions of ''q'p''(''a'') ≡ 0 (mod ''p'') with ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Root Modulo N
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo'' if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the ''Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. Elementary example The number 3 is a primitive root modulo 7 because :: \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
