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• Pseudoprime
  A '''pseudoprime''' is a [[composite number]] which passes some probabilistic [[primality test]]s. For example, a ''strong pseudoprime'' is a composite number that passes one iteration the [[Miller-Rabin pseudoprimality test]].
  1 KB (155 words) - 20:32, 25 July 2020
• Operation Billion Digits
  ...ibuted computing project]] that is searching for a "Billion Digit Mersenne prime". ...e also unfeasible because they require operations modulo the billion digit number. The only part of this project that can be undertaken today is [[trial fact
  6 KB (918 words) - 16:28, 24 July 2020
• M37
  | number=127411683030...973024694271 ...[[Roland Clarkson]], using [[Prime95]] written by [[George Woltman]]. The number is [http://www.mersenneforum.org/txt/37.txt {{Num|909526}} decimal digits]
  877 bytes (111 words) - 11:04, 18 February 2019
• 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
• Quadratic residue
  In [[mathematics]], a number {{V|q}} is called a '''quadratic residue''' [[modular arithmetic|modulo]] { ...w of quadratic reciprocity]] says something about quadratic residues and [[prime]]s.
  823 bytes (117 words) - 20:11, 26 October 2020
• 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 ...modulus is [[prime]]. Otherwise we can compute the square roots modulo the prime factors of <math>m</math> and then generate a solution using the Chinese Re
  5 KB (726 words) - 10:38, 6 February 2019
• Legendre symbol
  If <math>p</math> is an odd [[prime]] number and <math>a</math> is an [[integer]], then the Legendre symbol There are a number of useful properties of the Legendre symbol which can be used to speed up c
  2 KB (348 words) - 18:57, 28 September 2023
• 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
  1 KB (208 words) - 18:19, 2 October 2022
• Ten million digits
  ...llion [[decimal]] [[digit]]s) in June of 1999, the next [[EFF prizes]] for prime numbers was '''ten million decimal digits'''. ...was found, [[M46]]. By the end of 2010, all exponents that would produce a number less than {{Num|10000000}} digits had been [[primality test|tested]] at lea
  979 bytes (146 words) - 14:23, 6 March 2019
• Double Mersenne number
  ...efer to <math>2^{2^p-1}-1</math>. Early on it was thought that if M(p) was prime so too was MM(p). *MM(2) = <math>2^3-1</math> = 7, known prime since antiquity
  4 KB (655 words) - 14:50, 19 September 2021
• PSearch
  ...g project|distributed computing project]] in search of the largest [[Proth prime]]s. ! scope="col" | Number
  1 KB (182 words) - 08:17, 12 July 2020
• P-1 factorization method
  Let ''p'' be a prime which does not divide the integer ''a'', then <math>a^{p-1}\equiv 1 \mbox{( ...tiple of ''N'', so a [[greatest common divisor]] operation will reveal the prime divisor.
  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
• Self-initializing quadratic sieve
  Let N be the number to be factored. This number must not be a perfect power. If somehow we find two integers X and Y such t ...form <math>t^2 \equiv u\,\pmod N</math> where u is the product of small [[prime]] numbers. The set of these primes is the ''factor base''. These relations
  10 KB (1,763 words) - 02:56, 12 March 2019
• Lone Mersenne Hunters
  ...imeNet]] in order to eliminate [[Mersenne number]]s as possible [[Mersenne prime]] candidates. This work is suited to older and slower processors, often wit
  1 KB (213 words) - 09:58, 7 March 2019
• Fermat pseudoprimality test
  where ''p'' is a [[prime]] number and ''a'' is not multiple of <math>p</math>. ...t is not 1, the number must be composite. Otherwise the number is either a prime or a Fermat [[pseudoprime]] with respect to base <math>a</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: ...primality, and <math>N = 2^n\,k + 1</math> where <math>k</math> is an odd number.
  3 KB (432 words) - 15:33, 28 January 2019
• 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
• Generalized Fermat number
  There are different kinds of '''generalized [[Fermat number]]s'''. ...2^{2p^n}+2^{p^n}+1 \ = \ (2^{p^{n+1}}-1)/(2^{p^n}-1)</math> where p is the prime of apparition rank r (r(2)=1, r(3)=2, r(5)=3, ...) and n is greater or equa
  5 KB (774 words) - 07:39, 27 May 2024
• M51
  | number=148894445742...325217902591 '''M51''' normally refers to the 51st [[Mersenne prime]], in order of size from the smallest to greatest. This is the primary usag
  2 KB (255 words) - 05:53, 21 July 2021

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