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• Mersenne number
  A '''Mersenne number''' is a number of the form <math>2^n{-}1</math> where <math>n</math> is a non-negative [[i ...[prime]], it is called a [[Mersenne prime]], otherwise it is a [[composite number]].
  2 KB (351 words) - 11:28, 7 March 2019
• Fermat number
  In [[mathematics]], a '''Fermat number''', named after [[Pierre de Fermat]] who first studied them, is a positive ...ese factorisations can be found at [http://www.prothsearch.com/fermat.html Prime Factors of Fermat Numbers]
  12 KB (1,913 words) - 14:35, 9 August 2021
• Lucas-Lehmer test
  ...e last stage in the procedure employed by [[GIMPS]] for finding [[Mersenne prime]]s. Previous stages try to find factors, as explained on [[GIMPS factoring ...lete proof that this was not only true when p = 1 (mod 4), but for all odd prime exponents. The test therefore takes its name from the two mathematicians wh
  20 KB (3,572 words) - 14:30, 17 February 2019
• Mathematics
  '''Mathematics''' is the science of space, number and quantity. ...theorem: If you subtract an odd number from an even number you get an odd number.
  1 KB (186 words) - 17:00, 5 February 2019
• Fast Fourier transform
  Let ''x''₀, ...., ''x''_''n''-1 be [[complex number]]s. The DFT is defined by the formula ...lar misconception) there are O(''n'' log ''n'') FFTs for all ''n'', even [[prime]] ''n''.
  17 KB (2,684 words) - 18:50, 28 September 2023
• Perfect number
  In [[mathematics]], a '''perfect number''' is defined as an integer which is the sum of its proper positive divisor ...and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128.
  6 KB (885 words) - 11:33, 7 March 2019
• Primality test
  .... When the number is declared composite, the algorithm does not reveal the prime [[factor]]s. That is the job of the [[Factorization|factorization methods]] ...(which is far slower than a probable primality test except when the input number has a special form) is run on it.
  3 KB (501 words) - 05:20, 3 August 2021
• Primality testing program
  ...ne number]]s<br/>a &times; bⁿ±c (only factoring and [[probable prime|PRP]]-testing) | [[generalized Fermat number]]s
  2 KB (314 words) - 21:23, 29 August 2019
• Pépin's test
  '''Pépin's test''' is mainly used for proving the primality of [[Fermat number]]s, but it is of no help for finding the factors of such numbers. ...ams says: "The test that we today call Pépin's test is actually [[Proth's theorem|Proth's test]] with a proof provided by Lucas".
  2 KB (401 words) - 14:40, 6 March 2019
• Probable prime
  ...an [[integer]] that satisfies a specific condition also satisfied by all [[prime]] numbers.}} ...ecific conditions. While there may be probable primes that are [[Composite number|composite]] (called [[pseudoprime]]s), the condition is generally chosen in
  2 KB (232 words) - 07:28, 12 March 2024
• GIMPS factoring and sieving
  ...o do the Lucas-Lehmer Test; in fact, over 60% of [[Mersenne number]]s with prime exponents are eliminated from consideration as possible primes this way, so ...given Mersenne number up to some predetermined size, usually a prescribed number of bits.
  6 KB (962 words) - 10:08, 7 March 2019
• Modular square root
  A '''modular square root''' <math>r</math> of an [[integer]] number <math>a</math> modulo an integer <math>m</math> greater than 1 is an intege ...of <math>m</math> and then generate a solution using the Chinese Remainder Theorem.
  5 KB (726 words) - 10:38, 6 February 2019
• Law of quadratic reciprocity
  ...</math> is a [[quadratic residue]] or non-residue modulo another odd prime number <math>q</math> if we know whether <math>q</math> is a quadratic residue or This theorem was first proved by [[Carl Friedrich Gauss]] in 1801.
  1 KB (208 words) - 18:19, 2 October 2022
• P-1 factorization method
  It is based in the Fermat's Little Theorem that states that: Let ''p'' be a prime which does not divide the integer ''a'', then <math>a^{p-1}\equiv 1 \mbox{(
  5 KB (814 words) - 01:35, 12 March 2019
• Brent-Suyama extension
  ...of prime powers less than B1. Then by [[Fermat's Little Theorem]], a prime number p | S-1 if p-1 | E. ...tage 2 would then compute T=S^q = 3^E*q for successive prime q in the range (B1,B2]. Then p | T-1 if p-1 | q*E.
  2 KB (421 words) - 11:51, 28 January 2019
• Fermat pseudoprimality test
  The '''Fermat pseudoprimality test''' is based on the Fermat Little Theorem that states: where ''p'' is a [[prime]] number and ''a'' is not multiple of <math>p</math>.
  1 KB (164 words) - 10:56, 6 February 2019
• Miller-Rabin pseudoprimality test
  The '''Miller-Rabin pseudoprimality test''' is based in two facts for prime numbers: *The Fermat Little Theorem that states: <math>a^{p-1}\equiv 1\,\pmod p</math>.
  3 KB (432 words) - 15:33, 28 January 2019
• Why participate in GIMPS?
  ...s of the form 2^p-1, for some prime ''p'' (now called [[Mersenne prime|Mersennes]]). So the quest for these jewels began near 300 BC. ...umbers, how to characterize their [[factor]]s and discover those which are prime. In short, the tradition of seeking large primes (especially the Mersennes)
  7 KB (1,252 words) - 09:47, 7 March 2019
• Proth prime
  A '''Proth prime''' is not a true class of numbers, but primes in the form {{Kbn|+|k|n}} wit *[[Proth's theorem]]
  656 bytes (91 words) - 07:02, 31 August 2020
• What use are prime numbers anyway?
  ...Wikipedia:G. H. Hardy|Godfrey H. Hardy]] (1877 - 1947) said of his work in number theory :"Here is one science (number theory) at any rate whose very remoteness from ordinary human activities sh
  3 KB (497 words) - 07:17, 22 May 2020

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