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• Generalized Fermat prime
  ...'generalized Fermat prime''' is a [[generalized Fermat number]] which is [[prime]]. *[[Wikipedia:Fermat_number#Generalized_Fermat_primes|Generalized Fermat prime]]
  372 bytes (49 words) - 13:35, 6 March 2019
• PrimeGrid Generalized Fermat Prime Search
  ...t is searching for [[Generalized Fermat number#Dubner|Generalized Fermat]] prime numbers. ...hat any prime found would be larger than [[M51]], making it a world record prime number. If a new world record is found outside of PrimeGrid then this proje
  2 KB (310 words) - 06:38, 29 December 2023

Page text matches

• Marin Mersenne
  Mersenne is remembered today thanks to his association with the [[Mersenne prime]]s. The [[Mersenne twister]], named for him, is frequently used in computer ...for exchange of information. These include such notables as: [[Pierre de Fermat]], Pascal, Gassendi, Roberval, Beaugrand, Huygens, Pell, Galileo and Torric
  11 KB (1,582 words) - 01:17, 15 January 2024
• Fermat number
  In [[mathematics]], a '''Fermat number''', named after [[Pierre de Fermat]] who first studied them, is a positive integer of the form where {{Vn}} is a [[non-negative]] [[integer]]. The first eight Fermat numbers are (see {{OEIS|l|A000215}}):
  12 KB (1,913 words) - 14:35, 9 August 2021
• Derrick Henry Lehmer
  ...rsenne prime]]s <math>2^p{-}1</math>, extending its application to all odd prime exponents ''p'', and enabling him to add his name to the test so that it is ...called 'Lehmer Numbers'. He also clarified and extended Lucas' use of the Fermat congruence in primality testing, making its use widely known and in a paper
  6 KB (1,033 words) - 01:13, 15 January 2024
• Cunningham project
  ...the form <math>2kn+1</math>. Note that when the exponent <math>n</math> is prime, algebraic and Aurifeuillian factors are not possible, except for the trivi ...of the form <math>2kn+1</math>, <math>b \ge 2</math> and <math>n</math> is prime, except when <math>n</math> is a factor of <math>x-1</math>. In such cases,
  7 KB (1,150 words) - 23:48, 19 April 2023
• 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
• Mlucas
  ...hmer test]]s of prime-exponent [[Mersenne number]]s, and Pépin tests of [[Fermat number]]s. It is written by [[Ernst Mayer]] using C programming language an ...es not impose prize-sharing rules, should a user be lucky as to find a new prime eligible for the monetary prize offered by the [[Electronic Frontier Founda
  1 KB (198 words) - 07:28, 22 August 2019
• Conjectures 'R Us
  ...or a combination of algebraic and trivial factor(s), or make [[Generalized Fermat number]]'s. ...557 &lt; {{Vk}} &lt; 271129 has been extensively tested by the [[PrimeGrid Prime Sierpiński Problem]] and the [[PrimeGrid Extended Sierpiński Problem]] pr
  3 KB (503 words) - 04:44, 27 March 2024
• Elliptic curve method
  ...first six prime numbers multiplied together. If we multiply the first 2000 prime numbers together we get a huge number that has lots and lots of divisors. T ...at we work modulo a prime <math>p</math> (this will be an a-priori unknown prime factor of <math>N</math>). Given the set of points for which '''(1)''' hold
  19 KB (3,181 words) - 22:27, 6 July 2023
• Rho factorization method
  When N = pq where p and q are [[coprime]] but not necessarily [[prime]], we will see that after about <math>\sqrt p</math> elements the sequence ...s was invented by Richard Brent in 1980 who used it to factor the eighth [[Fermat number]].
  3 KB (558 words) - 10:28, 6 February 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]] ...the confidence grows, but we cannot be completely sure that the number is prime until a primality test (which is far slower than a probable primality test
  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. ...be used for proving the primality of other numbers, like the [[Generalized Fermat number]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1</math> with k = 5 instead of k
  2 KB (401 words) - 14:40, 6 March 2019
• Probable prime
  ...an [[integer]] that satisfies a specific condition also satisfied by all [[prime]] numbers.}} ...an [[integer]] that satisfies a specific condition also satisfied by all [[prime]] numbers. Different types of probable primes have different specific condi
  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 ...te numbers, it may make more sense to test the remaining 5% along with any prime potential factors by the power-mod algorithm than to do further sieving.
  6 KB (962 words) - 10:08, 7 March 2019
• LLR
  ...ith |{{V|c}}| ≠ 1 or {{V|k}} > {{Vb}}^{{Vn}} can be [[probable prime|PRP]]-tested. ...s and Wagstaff numbers (2^{{V|p}}+1)/3. The latter uses a strong Fermat PRP-test and the [[Vrba-Reix algorithm]].
  2 KB (300 words) - 22:00, 16 December 2023
• 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
  ...oduct 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
• GeneferCUDA
  ...DAGenefer''') is a program for finding large probable [[generalized Fermat prime]]s.
  362 bytes (48 words) - 09:20, 29 January 2019

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