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  • GIMPS chances of finding a prime number
    ...are used to calculate the probability of something happening based on the number of possible outcomes, not on what the last three or three hundred outcomes ...your next throw are 1:6. What has happened in the past does not affect the number of faces on the dice, which is all that is used to calculate the odds.
    3 KB (593 words) - 10:09, 7 March 2019

Page text matches

  • Williams prime
    A '''Williams number''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1. A '''Williams prime''' is a Williams number which is [[prime]].
    5 KB (744 words) - 07:30, 5 August 2019
  • Great Internet Mersenne Prime Search
    ...e Search: A [[distributed computing project]] for the search of [[Mersenne prime]]s.}} ...hat can be downloaded from the Internet, in order to search for [[Mersenne prime]] numbers.
    3 KB (450 words) - 14:37, 21 August 2019
  • List of known Mersenne primes
    *'''#''': number count of the Mersenne primes linked to that prime page *'''n-value''': value of exponent and if available link to [[The Prime Pages]] entry
    2 KB (360 words) - 09:44, 6 March 2019
  • Mersenne prime
    ...}. On the other hand, 15 = 16 − 1 = {{Kbn|4}}, for example, is not a prime, because 15 is divisible by 3 and 5. More generally, [[Mersenne number]]s (not necessarily primes, but candidates for primes) are numbers that are
    5 KB (857 words) - 14:53, 19 September 2021
  • 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
  • Landon Curt Noll
    ...ted in the New York times on 1978-11-21. The 18 year-olds were studying [[number theory]] at the time at CSUH with Dr. [[Derrick Henry Lehmer]] of [[Univers ...e [[multiplication]]s need in [[Lucas-Lehmer test]]ing of large [[Mersenne number]]s.
    2 KB (333 words) - 12:40, 9 February 2022
  • Laura A. Nickel
    ...l and Nickel were still high school students. For the verification of this number alone, the pair used almost eight hours of time running an assembly languag
    2 KB (254 words) - 01:23, 15 January 2024
  • 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
  • M50
    | number=467333183359...069762179071 '''M50''' normally refers to the 50th [[Mersenne prime]], in order of size from the smallest to greatest. This is the primary usag
    2 KB (333 words) - 13:16, 17 February 2019
  • M49
    | number=300376418084...391086436351 '''M49''' normally refers to the 49th [[Mersenne prime]], in order of size from the smallest to greatest. This is the primary usag
    2 KB (283 words) - 11:50, 18 February 2019
  • Derrick Henry Lehmer
    ...factoring a number ''N'' is hereby reduced to the discovery of an adequate number of quadratic residues ''R'' of ''N'' and the superposition of the correspon ...ber sieves]] to be run on a computer. He had previously built an automatic number sieve from a small electric motor and some bicycle chains hanging from spro
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Titanic prime
    A '''Titanic prime''' is a [[prime]] number whose decimal representation has {{Num|1000}} or more digits. The smallest titanic prime is {{T5000|58901|10<sup>999</sup>+7}}.
    394 bytes (48 words) - 11:40, 2 July 2020
  • Gigantic prime
    A '''gigantic prime''' is a [[prime]] number whose decimal representation has at least {{Num|10000}} [[digit]]s. The smallest gigantic prime is 10<sup>{{Num|9999}}</sup>+{{Num|33603}}.
    515 bytes (67 words) - 13:38, 6 March 2019
  • Megaprime
    A '''Megaprime''' is a [[prime]] number whose decimal representation has {{Num|1000000}} or more digits. There are ...st is avalable [http://primes.utm.edu/primes/search.php?MinDigits=1000000&&Number=10000&Style=HTML here].
    806 bytes (111 words) - 07:59, 14 July 2021
  • Gigaprime
    A '''Gigaprime''' is a [[prime]] number whose [[decimal]] representation has {{Num|1000000000}} or more [[digit]]s. [[Operation Billion Digits]] is factoring [[Mersenne number]]s in this range.
    871 bytes (119 words) - 07:54, 14 July 2021
  • Home prime
    ...me prime depends on the [[base]] (except in the case where ''n'' itself is prime). While it is expected that every ''n'' in every base has a home prime, experimental evidence indicates that these chains can get quite long.
    980 bytes (143 words) - 13:22, 6 March 2019
  • Cunningham project
    ...it is considered the oldest continuously ongoing activity in computational number theory. ...exponent. The second type is [[aurifeuillian factor]], in which the whole number can be split into two parts directly, for certain combination of values of
    7 KB (1,150 words) - 23:48, 19 April 2023
  • M25
    | number=448679166119...353511882751 The 25th [[Mersenne prime]], in order from smallest to largest and in order of discovery.
    2 KB (303 words) - 11:01, 26 February 2019
  • 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
  • Édouard Lucas
    ...[[Mersenne prime]] for almost 75 years, and is still the highest [[prime]] number discovered without the aid of a computer.
    2 KB (296 words) - 01:09, 15 January 2024
  • Edson Smith
    .... In August 2008, one of these computers found a [[M47| World record prime number.]] Since the first [[Mersenne prime]] found by a computer ([[M13]]) was found at UCLA (as were 6 others in the
    4 KB (564 words) - 00:11, 15 January 2024
  • 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
  • Factorial number
    ...or bang) after a number, it represents multiplying a number by all [[whole number|whole numbers]] smaller than it. *[[Factorial prime]]
    729 bytes (93 words) - 13:40, 5 November 2023
  • Factor
    A '''factor''' is one of the numbers or expressions that make up another number by [[multiplication]]. Let a and b be integers. Then a divides b (which may ...a number that has factors other than itself and 1 is called a [[composite number]].
    576 bytes (107 words) - 19:03, 5 February 2019
  • Composite number
    A positive [[integer]] is '''composite''' if it is neither [[prime]] nor equal to 1. The smallest composite is 4. ...he integers <math>a</math> and <math>b</math> are both greater than 1, the number is composite.
    358 bytes (56 words) - 23:30, 26 October 2020
  • Factoring Database
    **[[Home prime]]s of various bases **Greatest prime factor ^2+1, ^2+2, ^2-1, ^2-2, ^3+1, ^3-1
    1 KB (144 words) - 13:44, 24 January 2019
  • Factorization
    '''Factorization''' is the process of finding [[prime]] [[factor]]s. This article will only cover integer factorization. ...t can be seen that we have to proceed recursively in order to find all the prime factors of ''c''.
    4 KB (642 words) - 12:57, 5 March 2019
  • EFF prizes
    ...substantial award for the person that discovers a ten million digit prime number. If you find such a prime with the software provided, GIMPS will claim the award and distribute the a
    2 KB (321 words) - 18:50, 14 December 2023
  • Mlucas
    ...st]]s of prime-exponent [[Mersenne number]]s, and Pépin tests of [[Fermat number]]s. It is written by [[Ernst Mayer]] using C programming language and [[ARM ...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
  • M48
    | number=581887266232...071724285951 '''M48''' normally refers to the 48th [[Mersenne prime]], in order of size from the smallest to greatest. This is the primary usag
    2 KB (235 words) - 11:49, 18 February 2019
  • M47
    | number=316470269330...166697152511 ...] article. The number now refered to as M47 was actually the 45th Mersenne prime found. [[M45]], [[M46]], and M47 were discovered in the order of M47, M45 (
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    ...], a [[distributed computing]] project dedicated to finding new [[Mersenne prime]] numbers. More specifically, Prime95 refers to the Windows and Mac OS X ve ...Awards]). As such, a user who uses Prime95 to discover a qualifying prime number would not be able to claim the prize directly. A free software package woul
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Odd number
    An '''odd number''' is any [[integer]] that is not divisible by 2. ...expressed in decimal notation, the odd numbers end in 1, 3, 5, 7 or 9. All prime numbers except 2 are odd.
    316 bytes (42 words) - 11:21, 7 March 2019
  • M41
    | number=299410429404...882733969407 '''M41''' is the short hand used to refer to the 41st [[Mersenne prime]] 2<sup>{{Num|24036583}}</sup>-1.
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    | number=315416475618...411652943871 '''M43''' is the short hand used to refer to the 43rd [[Mersenne prime]] 2<sup>{{Num|30402457}}</sup>-1.
    1 KB (191 words) - 11:31, 18 February 2019
  • M46
    | number=169873516452...765562314751 '''M46''' is the short hand used to refer to the 46th [[Mersenne prime]] 2<sup>{{Num|42643801}}</sup>-1.
    2 KB (248 words) - 11:45, 18 February 2019
  • Distributed computing
    There are a number of individuals or groups in the fields of science, mathematics, cryptograph ...ernet, to look for extra-terrestrial radio signals, to look for [[Mersenne prime]]s so large that they have more than [[ten million digits]], to find more e
    4 KB (674 words) - 12:11, 19 February 2019
  • M42
    | number=122164630061...280577077247 '''M42''' refers to the 42nd [[Mersenne prime]] 2<sup>{{Num|25964951}}</sup>-1.
    934 bytes (118 words) - 11:26, 18 February 2019
  • M1
    | number= 3 [[Category:Mersenne prime|M01]]
    193 bytes (19 words) - 13:43, 17 February 2019
  • Conjectures 'R Us
    ...project]] in search for lowest [[Sierpiński number|Sierpiński]]/[[Riesel number|Riesel]] values.}} ...mbination of algebraic and trivial factor(s), or make [[Generalized Fermat number]]'s.
    3 KB (503 words) - 02:20, 1 May 2024
  • Fast Fourier transform
    Let ''x''<sub>0</sub>, ...., ''x''<sub>''n''-1</sub> 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
  • GpuLucas
    ...[[CUDA]]-based program written by [[Andrew Thall]] for testing [[Mersenne number]]s for primality. ...l, A. [http://andrewthall.org/papers/gpuMersenne2011MKII.pdf Fast Mersenne Prime Testing on the GPU] (2011)
    2 KB (239 words) - 11:12, 13 February 2019
  • GpuOwL
    {{InfoboxProgram|workload=[[Lucas-Lehmer test|LL]], [[Probable prime|PRP]]|title=gpuOwL|release=2017|latest=7.2<br>2020-11-01}} ...s a [[OpenCL]]-based program written by Mihai Preda for testing [[Mersenne number]]s for primality.
    1 KB (216 words) - 05:22, 1 December 2020
  • MersenneForum
    ...upport [[GIMPS]], the broader community of [[Mersenne number]]s, [[prime]] number, and factoring projects. In addition to being the de facto help and support ==Prime number software discussion and development==
    2 KB (293 words) - 17:33, 5 July 2019
  • Primo
    '''Primo''' is a computer program which tests numbers for [[prime|primality]] using the [[Elliptic Curve Primality Proving]] (ECPP) [[algorit ...ot require a number to be of any specific form. If a number is found to be prime, a [[primality certificate]] is produced, which can be quickly verified.
    1 KB (191 words) - 20:33, 12 May 2020
  • M32
    | number=174135906820...328544677887 The '''32nd [[Mersenne prime]]''', both in size (smallest to largest) and in order of discover.
    2 KB (279 words) - 08:35, 18 February 2019
  • M33
    | number=129498125604...243500142591 '''M33''' refers to 33rd [[Mersenne prime]] number 2<sup>{{Num|859433}}</sup>-1.
    814 bytes (97 words) - 08:38, 18 February 2019
  • M34
    | number=412245773621...976089366527 ...ber {{Num|378632}} [[decimal]] [[digit]]s long. The number was found to be prime in 1996.
    3 KB (513 words) - 08:42, 18 February 2019
  • Rational number
    A '''rational number''' is a [[real number]] which can be written as <math>\frac{a}{b}</math> or <math>a/b</math> wher ...r [[greatest common divisor]]. This operation does not change the rational number represented by the fraction.
    3 KB (541 words) - 15:01, 26 March 2023
  • Binary
    ...iness' of Mersenne numbers makes calculations in the search for [[Mersenne prime]]s a bit easier.
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...[[Mersenne number]]s are repunit ('''rep'''eated '''unit''', "1" being the number referred to as "unity") numbers. 111 is a repunit, in base 2 it is equal to A '''Repunit prime''' is a repunit which is also [[prime]].
    1 KB (207 words) - 08:04, 12 March 2024
  • Trial factoring
    ...are found, the number in question is prime; otherwise, it is a [[composite number]]. ..., P(2) = 3, P(3) = 5, etc, then the last prime factor possibility for some number N would be P(m) where P(m + 1) squared exceeds N.
    7 KB (1,221 words) - 13:20, 11 February 2019
  • M40
    | number=125976895450...762855682047 ...Num|6320430}} decimal digits] long. This prime number was the sixth record prime found by the [[GIMPS]] project.
    1 KB (189 words) - 11:17, 18 February 2019
  • M39
    | number=924947738006...470256259071 ...[[Michael Cameron]], using [[Prime95]] written by [[George Woltman]]. The number is [http://www.mersenneforum.org/txt/39.txt {{Num|4053946}} decimal digits]
    868 bytes (109 words) - 11:14, 18 February 2019
  • M44
    | number=124575026015...154053967871 '''M44''' is the short hand used to refer to the 44th [[Mersenne prime]]. Currently that designation belongs to 2<sup>{{Num|32582657}}</sup>-1.
    997 bytes (129 words) - 11:35, 18 February 2019
  • M45
    | number=202254406890...022308220927 ...''' normally refers to 2<sup>{{Num|37156667}}</sup>-1, the 45th [[Mersenne prime]] in order of size from the smallest to greatest. This is the primary usage
    2 KB (251 words) - 11:40, 18 February 2019
  • Richard Crandall
    ...uter]] scientist and physicist who has made contributions to computational number theory. He received a doctorate from [[Massachusetts Institute of Technolog His Erdös number is 2. He was one of the primary verifiers of [[M32]], [[M33]], and [[M34]].
    3 KB (431 words) - 11:36, 14 January 2024
  • M38
    | number=437075744127...142924193791 ...[[Nayan Hajratwala]], using [[Prime95]] written by [[George Woltman]]. The number is [http://www.mersenneforum.org/txt/38.txt {{Num|2098960}} decimal digits]
    1 KB (165 words) - 11:10, 18 February 2019
  • Nayan Hajratwala
    ...houseCoopers employee from Michigan who discovered the [[M38|38th Mersenne prime]], 2<sup>{{Num|6972593}}</sup>-1. ...99-06-01, Hajratwala's 350 MHz IBM Aptiva home computer first reported the prime to the [[GIMPS]] server. The computer had taken 111 days to complete the te
    809 bytes (109 words) - 23:55, 14 January 2024
  • Irrational base discrete weighted transform
    ...T''') is a variant of the [[Fast Fourier transform]] using an [[Irrational number|irrational]] base. It was proposed by [[Richard Crandall]] and [[Barry Fagi The IBDWT is used to perform FFT multiplication modulo [[Mersenne number]] in such programs as [[Prime95]], [[CUDALucas]], [[Glucas]], [[gpuLucas]].
    1 KB (172 words) - 18:49, 28 September 2023
  • PrimeKit
    ...Mathematica implementations of all 112 algorithms discussed in the book ''Prime Numbers: A Computational Perspective'' (2001) by [[Richard Crandall]] and C ...optimized), but there is also an "Extras" folder containing some efficient number-theoretical C sources.
    1 KB (125 words) - 09:38, 23 January 2019
  • Coprime
    ...h>\gcd{(x,y)} = 1</math>). This does not mean that any of these numbers is prime. :Two random numbers are coprime with a probability over 60% (the exact number is <math>6/\pi^2</math>).
    738 bytes (112 words) - 09:50, 23 January 2019
  • Greatest common divisor
    ...re <math>a</math> and <math>b</math> are positive integers, is the maximum number that divides both <math>a</math> and <math>b</math>. ...] or relatively prime. This does not mean that either of these numbers are prime.
    2 KB (339 words) - 18:38, 27 September 2023
  • Elliptic curve method
    ...le of a point on a random elliptic curve [[modular arithmetic|modulo]] the number to be factored. It is currently the best [[algorithm]] known, among those w ...ber]]. This method cannot be used when it is not known in advance that the number is composite, so it cannot be used as a [[primality test]].
    19 KB (3,181 words) - 22:27, 6 July 2023
  • M35
    | number=814717564412...868451315711 '''M35''' is the 35th [[Mersenne prime]], both in order of size and date of discovery.
    2 KB (224 words) - 11:00, 18 February 2019
  • University of California, Los Angeles
    As an institution, UCLA has contributed to the discovery of 8 [[Mersenne prime]]s. This is more than any other university. [[University of Central Missour ...number of digits of the largest known [[prime]] (in general) and Mersenne Prime from 79 and 39 (respectively) to 687.
    2 KB (347 words) - 14:54, 19 September 2021
  • Raphael M. Robinson
    ...0) proved that an essentially undecidable theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic ''Q''. '' ..., [[M16|2203]], [[M17|2281]]. He discovered the last 5 of these [[Mersenne prime]]s, the largest ones known at the time.
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    | number=623340076248...743729201151 ...It took Spence's 100 MHz [[Pentium]] computer 15 days to prove the number prime. Alan White Managing Director at Technology Business Solutions, who provide
    2 KB (279 words) - 11:01, 18 February 2019
  • Jonathan Pace
    ...l engineer. He is credited with discovery of the [[M50|50th known Mersenne prime]] {{Kbn|77232917}}. ...en he read an article about the discovery of the [[M40|40th known Mersenne prime]].
    2 KB (242 words) - 00:08, 15 January 2024
  • Sierpiński problem
    The '''Sierpiński problem''' in [[number theory]] was proposed by [[Wacław Sierpiński]] in 1960. ...[[composite number]] {{V|N}}, then {{Vk}} is said to be a '''[[Sierpiński number]]'''.
    5 KB (650 words) - 10:25, 26 March 2024
  • Seventeen or Bust
    ...(SOB)''' was a [[distributed computing]] project working on a problem in [[number theory]] called the [[Sierpiński problem]]. It is currently a subproject o ...we are running [[primality test]]s called [[Probable prime|PRP]] (Probable Prime), which take a very long time, for every candidate in the queue.
    3 KB (544 words) - 16:44, 21 July 2019
  • M2
    | number= 7 [[Category:Mersenne prime|M02]]
    193 bytes (19 words) - 13:43, 17 February 2019
  • M3
    | number= 31 [[Category:Mersenne prime|M03]]
    194 bytes (19 words) - 13:43, 17 February 2019
  • M4
    | number= 127 [[Category:Mersenne prime|M04]]
    195 bytes (19 words) - 13:44, 17 February 2019
  • M5
    | number= 8191 [[Category:Mersenne prime|M05]]
    204 bytes (18 words) - 13:46, 17 February 2019
  • 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
  • M9
    | number=2305843009213693951 The ninth [[Mersenne prime]], 2<sup>61</sup>-1 or {{Num|2305843009213693951}}.
    2 KB (213 words) - 14:30, 17 February 2019
  • Double check
    *human error (entering wrong number to test, misreading data, etc.) ...t]] does a verfication on all [[factor]]s reported. (It is easy to check a number for a single factor.)
    2 KB (373 words) - 15:08, 5 June 2019
  • Residue
    ...ctly divisible. For the L-L test a zero residue means that the number is [[prime]]. ...test to produced matching erroneaous residues (meaning they both missed a prime) out of a pool of ~ 18.4 pentillion numbers, this is considered to be impos
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    The [[Great Internet Mersenne Prime Search]] (GIMPS) as a project is based on two related items: theory and pra ...very large Mersenne prime candidates to be tested for primality ("is it a prime?") faster than other sorts of would-be primes of the same magnitude. "Faste
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...ics cards, this is a very fast program. The name mfaktc is "'''M'''ersenne number '''fakt'''oring with '''C'''UDA", it is a mixture of English with the Germa *Prime exponents between 100000 and <math>2^{32}-1</math>
    5 KB (765 words) - 14:54, 25 February 2019
  • Rho factorization method
    The idea is to create a sequence iterating a polynomial modulo the number to be factored. 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
    3 KB (558 words) - 10:28, 6 February 2019
  • Riesel number
    ...value of ''k'' such that {{Kbn|k|n}} is always composite for all [[natural number]]s. ...Sierpiński problem]] article, [[Hans Riesel]] found in 1956 that [[Riesel prime 2 509203|{{Kbn|509203|n}}]] is always composite.
    827 bytes (112 words) - 08:21, 25 March 2024
  • M12
    | number=170141183460...715884105727 ...n a "smart phone" in under one second. This was the largest known Mersenne prime until 1952, when [[Raphael M. Robinson|Robinson]] at [[University of Califo
    2 KB (354 words) - 14:52, 19 September 2021
  • 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
  • Lucas primality test
    ...' invented in 1891 by [[Édouard Lucas]], determines whether a number N is prime or not, using the complete factorization of N-1. ...is not congruent to 1 modulo N for any prime divisor q of N-1, then N is a prime.
    1 KB (177 words) - 14:31, 17 February 2019
  • Primality testing program
    ...ne number]]s<br/>a &times; b<sup>n</sup>±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. ...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 = 3.
    2 KB (401 words) - 14:40, 6 March 2019
  • Mprime (Cray)
    ...later versions with [[Paul Gage]]), for testing [[Mersenne number]]s for [[Prime|primality]] on [[Cray Research|Cray]] [[Classes of computers#Supercomputer| This software is responsible for the discovery of 7 [[Mersenne prime]]s. It used [[Fast Fourier transform]]s for the [[multiplication]] of very
    639 bytes (92 words) - 12:02, 7 February 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
  • 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
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  • 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
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  • 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
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  • 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
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  • PSearch
    ...g project|distributed computing project]] in search of the largest [[Proth prime]]s. ! scope="col" | Number
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  • 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.
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  • 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<sup>q</sup> = 3<sup>E*q</sup> for successive prime q in the range (B1,B2]. Then p | T-1 if p-1 | q*E.
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  • 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
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  • 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
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  • 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>.
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  • 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.
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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]]
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  • 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
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  • 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
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  • Patrick Laroche
    ...less than 4 months and on just his fourth try, he discovered the new prime number. By way of comparison, some GIMPS participants have searched for more than He is credited with discovery of the [[M51|51th known Mersenne prime]] 2<sup>{{Num|82589933}}</sup>-1.
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  • PrimeGrid
    '''PrimeGrid''' is a [[distributed computing]] project for searching for [[prime]] numbers of world-record size. It makes use of the [[BOINC|Berkeley Open I :[[PrimeGrid 321 Prime Search|321 Prime Search]] searching for mega primes of the form {{Kbn|±|3|2|n}}.
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  • Worktype
    ...nne primes have been missed, and lastly finding [[factor]]s for [[Mersenne number]]s). ...ber. A found factor will conclusively prove that the number is [[Composite number|composite]], which eliminates the need to run a [[primality test]].
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  • Sieve of Eratosthenes
    The '''sieve of Eratosthenes''' is a method to find all [[prime]] numbers smaller than a given integer <math>N</math>. It's invention is cr ...h>, then <math>N</math> is [[composite number|composite]]; otherwise it is prime.
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  • Aliquot sequence
    An '''aliquot sequence''' is a sequence of numbers generated from an initial number using the sigma <math>\sigma(n)</math> function. ...visors''' of the number, <math>n</math>, which are all the divisors of the number, excluding itself. Therefore, sequences are generated thusly:
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  • Prime
    ...ger]] greater than 1 that is only divisible by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. ...that the idea of a 'largest prime' is fiction. And, if there is no largest prime the primes must be infinite.
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  • Twin prime
    ...mber]] that differs from another prime number by two, for example the twin prime pair (41, 43). ==Count of twin prime pairs==
    2 KB (255 words) - 06:08, 21 February 2023
  • Riesel Prime Search
    '''Riesel Prime Search''' (RPS) is a prime searching project established in 2005 by [[Predrag Minovic]] (Kosmaj). :The project is searching for [[Riesel prime]]s {{Kbn|k|n}}, {{Vk}} > 1.
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  • Nomenclature and notation
    ==[[Mersenne number]]== ...{p}-1</math> is generally used when discussing the search for a [[Mersenne prime]].
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  • Math rendering
    ...ame consisting of letters only. Command names are terminated by a space, a number or any other "non-letter". | <code>\prime, f^\prime, f', f'', f^{(3)}, \dot y, \ddot y</code>
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  • Why participate in GIMPS?
    ...s of the form 2<sup>p</sup>-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)
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  • Sieving
    '''Sieving''' is an algorithm to discover [[smooth number]]s and [[prime]] numbers from a sequence of [[integer]]s much faster than [[trial factorin The next step depends on whether we need to find prime number or smooth numbers.
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  • Smooth number
    A '''smooth number''' is an [[integer]] whose [[prime]] [[factor]]s are less or equal to a prescribed bound ('''smoothness bound' If this bound is B, we can say that the number is B-smooth.
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  • Sieving program
    *[[MultiSieve]] (performing sieving of different kinds of number) http://home.roadrunner.com/~mrodenkirch/home/MultiSieve.html ...ng [[twin prime]]s of the same form) http://sites.google.com/site/kenscode/prime-programs
    2 KB (220 words) - 11:42, 7 March 2019
  • NewPGen
    *k&times;b<sup>n</sup>±1 ([[twin prime]]s) *k&times;b<sup>n</sup>-1, 2k&times;b<sup>n</sup>-1 ([[Sophie Germain prime|Sophie Germain]])
    3 KB (529 words) - 09:32, 7 March 2019
  • Sophie Germain prime
    ..., 23 is a Sophie Germain prime because it is a prime and 2*23+1 = 47, also prime. ...e: Possible to give the No. (if available) in the comments of the [[Riesel prime]] page (see [[:Template:NVal|Template NVal]])?
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  • M13
    | number=686479766013...291115057151 '''M13''' is the 13th known [[Mersenne prime]] 2<sup>521</sup>-1 found on 1952-01-30 by [[Raphael M. Robinson]].
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  • M14
    | number=531137992816...219031728127 '''M14''' is the 14th known [[Mersenne prime]] 2<sup>607</sup>-1 found on 1952-01-30 by [[Raphael M. Robinson]].
    420 bytes (48 words) - 14:49, 19 September 2021
  • M15
    | number=104079321946...703168729087 '''M15''' is the 15th known [[Mersenne prime]] 2<sup>{{Num|1279}}</sup>-1 found on 1952-06-25 by [[Raphael M. Robinson]]
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  • M16
    | number=147597991521...686697771007 '''M16''' is the 16th known [[Mersenne prime]] 2<sup>{{Num|2203}}</sup>-1 found on 1952-10-07 by [[Raphael M. Robinson]]
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  • M17
    | number=446087557183...418132836351 '''M17''' is the 17th known [[Mersenne prime]] 2<sup>{{Num|2281}}</sup>-1 found on 1952-11-09 by [[Raphael M. Robinson]]
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  • M18
    | number=259117086013...362909315071 '''M18''' is the 18th known [[Mersenne prime]] 2<sup>{{Num|3217}}</sup>-1 found on 1957-09-08 by [[Hans Riesel]].
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  • M11
    | number=162259276829...578010288127 '''M11''' is the 11th known [[Mersenne prime]] 2<sup>107</sup>-1</math> found in 1914 by [[Ralph Ernest Powers]].
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  • M10
    | number=618970019642...137449562111 '''M10''' is the 10th known [[Mersenne prime]] <math>2^{89}-1</math> found in 1911 by [[Ralph Ernest Powers]].
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  • Proth prime
    A '''Proth prime''' is not a true class of numbers, but primes in the form {{Kbn|+|k|n}} wit *[[Proth prime table|Table]] with all available ''k''-values
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  • Riesel prime
    Although there's no official definition of a '''Riesel prime''' mostly all primes of the form {{Kbn|k|n}} with 2<sup>{{Vn}}</sup> > {{Vk *IDs and found dates from the [[The Prime Pages]]
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  • M26
    | number=402874115778...523779264511 '''M26''' is the 26th known [[Mersenne prime]] 2<sup>{{Num|23209}}</sup>-1 found on 1979-02-09 by [[Landon Curt Noll]].
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  • Introductory stuff
    It is feasible, but unlikely. A [[positive claim]], that of a new [[Mersenne prime]], is subject to [[Double check|double]] and [[triple check]]s by others, i ==What is a Mersenne prime?==
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  • 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
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  • Glossary
    *'''[[Composite number]]''' - An [[integer]] that is not [[prime]]. *'''[[Fermat number]]''' - Numbers of the form <math>2^{2^n} + 1</math>.
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  • Pocklington's theorem
    ...vented by H. C. Pocklington in 1914, which is a [[primality test]] for the number ''N'', states: ...v\,1\,\pmod{n}</math> and <math>gcd(a^{(N-1)/q}-1,N) = 1</math>, then each prime factor ''q'' of ''N'' has the form <math>q^kr+1</math>.
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  • Multiple polynomial quadratic sieve
    Let <math>N</math> be the number to be factored. This number must not be a perfect power. If somehow we find two integers <math>X</math> ...^2 \equiv u\,\pmod N</math> where <math>u</math> is the product of small [[prime]] numbers. The set of these primes is the ''factor base''. These relations
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  • Proth number
    In [[number theory]], a '''Proth number''' is a number of the form A [[Proth prime]] is a Proth number, which is prime.
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  • 27121 Search
    ...[[PrimeGrid]], is a [[Distributed computing]] project, which searches for prime numbers of the form: 27 &times; 2<sup>n</sup> ± 1 and 121 &times; 2<sup>n< ...tion system, which makes it easy to use [[PRPclient]] to reserve a testing number directly from [http://prpnet.primegrid.com:12006/ the website].
    983 bytes (138 words) - 13:25, 8 February 2019
  • 12121 Search
    ...121 Search''' is a [[Distributed computing]] project, which searches for [[prime]]s of the form: 121 &times; 2<sup>n</sup>-1, it also searches for primes of ...tion system, which makes it easy to use [[PRPclient]] to reserve a testing number directly from [http://prpnet.primegrid.com:12001/ the website].
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  • Assignments
    ...int files, results.txt file and the prime.spl, if it exists. Be aware that prime.spl is written to while an exponent is in progress and not just after compl ...ly large quantity of exponents, add this line to your [[GIMPS client files|prime.ini]] file:
    20 KB (3,473 words) - 18:42, 14 December 2023
  • GIMPS type of work
    ==Factoring of a prime number (exponent) candidate== ...this when [[mfaktc]]/[[mfakto]] are used. There is no chance of finding a prime through factoring.
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  • GIMPS networking, PrimeNet and communication
    prime.ini prime.spl
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  • GIMPS chances of finding a prime number
    ...are used to calculate the probability of something happening based on the number of possible outcomes, not on what the last three or three hundred outcomes ...your next throw are 1:6. What has happened in the past does not affect the number of faces on the dice, which is all that is used to calculate the odds.
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  • Worktodo.txt
    :AdvancedTest simply [[Lucas-Lehmer test|LL]] tests the given [[Mersenne number]], (ignoring any sort of prefactoring) and is used by [[Prime95]] when you ;Probable prime
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  • Assignment ID
    ...ponent) PrimeNet will issue a 32 character hexadecimal assignment ID. This number appears to be random (at least in part), so an individual cannot construct [[Category:Great Internet Mersenne Prime Search]]
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  • Triple check
    ...composite]], (this assumes that the residue is not zero, otherwise it is [[prime]]). ...ainst some error in either software or hardware design or manufacture, the number will be tested using [[Mlucas]] or [[Glucas]] on a computer using an Itaniu
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  • 79.3 million
    ...assignment/testing to a maximum of ~'''79.3 million'''. This seemingly odd number is derived from the fact that it is the largest to practically test using a [[Category:Great Internet Mersenne Prime Search]]
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  • Strong law of small numbers
    ...s after [[M12]] was proven prime). Many believe that the [[Double Mersenne number#Catalan Sequence|Catalan Sequence]] is also a case of this law, since only Because relatively few [[Mersenne prime]]s are known, people often conjecture about patterns.
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  • P+1 factorization method
    If we define <math>D = u^2 - 4</math>, then for any odd prime <math>p</math>, <math>p</math> divides both gcd(<math>N</math>, <math>U_M</ ...nt of p-1. So when the [[greatest common divisor|gcd]] does not reveal the prime factor of <math>N</math> we retry with a different value of <math>u</math>
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  • Woodall number
    In [[number theory]], a '''Woodall number''' W<sub>n</sub> is any [[natural number]] of the form for some natural number ''n''.
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  • Gerbicz error checking
    ...ts]], the technique is used to ensure validity of PRP tests for [[Mersenne number]]s: ...e programs and [[PRST]] in an extended version for PRP tests on additional number forms.
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  • ECPP-DJ
    ...aic%20number%20theory%20-%20Cohen.pdf "A Course in Computational Algebraic Number Theory"] (1993). The ECM factoring and manipulation was heavily inspired by :PRIME
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  • M19
    | number=190797007524...815350484991 [[Category:Mersenne prime]]
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  • M20
    | number=285542542228...902608580607 [[Category:Mersenne prime]]
    280 bytes (23 words) - 22:33, 17 February 2019
  • M21
    | number=478220278805...826225754111 [[Category:Mersenne prime]]
    281 bytes (25 words) - 22:35, 17 February 2019
  • M22
    | number=346088282490...883789463551 [[Category:Mersenne prime]]
    281 bytes (25 words) - 22:37, 17 February 2019
  • M23
    | number=281411201369...087696392191 [[Category:Mersenne prime]]
    282 bytes (25 words) - 22:38, 17 February 2019
  • M24
    | number=431542479738...030968041471 [[Category:Mersenne prime]]
    283 bytes (24 words) - 22:48, 17 February 2019
  • M8
    | number=2147483647 [[Category:Mersenne prime|M08]]
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  • M7
    | number=524287 [[Category:Mersenne prime|M07]]
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  • M6
    | number=131071 [[Category:Mersenne prime|M06]]
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  • GIMPS PrimeNet reports
    ...1000 most recently 'cleared' exponents (those determined to be [[composite number]]s). A new list is generate each hour at the start of the hour. This list a ...of 2008. Provides a "classic" summary of the search status for [[Mersenne number]]s with exponents below [[79.3 million]] broken down by [[Fast Fourier tran
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  • 15k Search
    The first goal is, to spend as little CPU time as possible, per [[titanic prime]] found. The second goal is, to find a [[prime]] in a secure manner, worthy of an [[EFF prizes|E.F.F. prize]].
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  • M27
    | number=854509824303...961011228671 [[Category:Mersenne prime]]
    302 bytes (26 words) - 08:16, 18 February 2019
  • M28
    | number=536927995502...709433438207 [[Category:Mersenne prime]]
    278 bytes (23 words) - 08:22, 18 February 2019
  • M29
    | number=521928313341...083465515007 [[Category:Mersenne prime]]
    296 bytes (26 words) - 08:25, 18 February 2019
  • M30
    | number=512740276269...455730061311 [[Category:Mersenne prime]]
    282 bytes (24 words) - 08:28, 18 February 2019
  • M31
    | number=746093103064...103815528447 [[Category:Mersenne prime]]
    285 bytes (24 words) - 08:30, 18 February 2019
  • Milestones report
    *'''Discovery of new [[Mersenne prime]]s.''' *[[Double check]]ing all numbers less than a given Mersenne Prime, thus proving its place in the sequence of MP's (e.g. 'verifying' that M32
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  • PrimeNet summary report
    ...of the active machines. Since not all machines are contributing 24/7 this number is not close to 100%. [[Computing power|GHz-days]] and TFLOPS are used. ...in during the period. For the 30 day period, this is higher than the total number in the 'Resources Registered' section, because many assginments take much l
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  • GIMPS milestones
    ...nternet Mersenne Prime Search]] has already discovered thirteen [[Mersenne prime]]s! | 2009-04-12 || '''Prime [[M46]] = M(42643801) discovered!!'''
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  • Riesel problem 1
    The '''Riesel problem''' involves determining the smallest [[Riesel number]]. ...that {{Kbn|k|2|n}} is not prime for any integer {{Vn}}. He showed that the number {{Vk}} = ''{{Num|509203}}'' has this property.
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  • Frequently suggested improvements
    ...reduction. Since we square at each step of the [[Lucas-Lehmer test]], the number of digits approximately doubles each time. So after only 50 iterations, the ...ble amount of elapsed time (when [[double check]]ing a supposed [[Mersenne prime]]). In this case SMP variations of the LL test can be used and are used - s
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  • LL test distribution
    At any given instant, most of [[Great Internet Mersenne Prime Search|GIMPS]]'s LL tests fall roughly into one of 2 categories. As time pa ...ent first time [[Lucas-Lehmer test|LL tests]] are taking place. [[Mersenne prime]] discoveries almost always take place in this range. As of Feb. 2020, most
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  • Mersenne.ca
    ...ake a large amount of RAM, and have no possibility of finding a [[Mersenne prime]], many GIMPS users purposely skipped the P-1 stage to spend more time doin ...cached for quick access to the "worst" ranges of exponents, where large a number of exponents have been poorly factored. Most options are user-configurable
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  • Cullen number
    A '''Cullen number''' {{V|C<sub>n</sub>}} is a number of the form {{Kbn|+|n|2|n}}, a '''generalized Cullen number''' base {{Vb}} is a number of the form {{Kbn|+|n|b|n}}. '''(perhaps own page?)'''
    2 KB (252 words) - 17:39, 31 August 2021
  • Mtsieve
    *mfsieve: search for factors of [[Multifactorial number]]s *cksieve: search for factors of [[Carol-Kynea prime]]s
    2 KB (338 words) - 06:58, 28 March 2023
  • Available Double-Triple checks
    ...re listed the available [[worktype|double or triple checks]] of [[Mersenne number]]s. [[Category:Great Internet Mersenne Prime Search]]
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  • Carol-Kynea prime
    ...rdinated in a [[MersenneForum]] thread, and the results are stored here on Prime-Wiki. ...h>. A Carol/Kynea prime is a [[prime]] which has one of the above forms. A prime of these forms must satisfy the following criteria:
    8 KB (1,172 words) - 00:38, 6 July 2023
  • Multifactorial prime
    A '''Multifactorial prime''' is a [[Multifactorial number]] which is prime and of the form <math>\ n!_2{±}1,\ n!_3{±}1,\ n!_4{±}1</math>, and so on *[[Factorial number]]
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  • Multifactorial number
    A [[Factorial number]] is defined by the product A '''Multifactorial number''' is denoted by
    560 bytes (81 words) - 14:36, 20 July 2021
  • Riesel 2 Low-weight
    include={Riesel prime}:Rk,{Riesel prime}:RCount,{Riesel prime}:RNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
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  • Factorial prime
    A '''Factorial prime''' is a [[prime]] of the form '''[[Factorial number]] ± 1'''. *[[Multifactorial prime]]
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  • Williams prime MM least
    ...or any base {{Vb}} with 2 ≤ {{Vb}} ≤ 2049 which generates a [[Williams prime]] of the form {{Kbn|(b-1)|b|n}}. {{HistF|2019-06-05|23089 ([[Williams prime MM 1801|{{Vb}}=1801]])|Karsten Bonath}}
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  • Proth prime 2 1
    {{Proth prime |PRemarks=These n-values form the [[Fermat number|Fermat primes]].
    212 bytes (30 words) - 15:35, 2 October 2022
  • Proth prime 2 9
    {{Proth prime ...}}<br>For all even {{Vn}}-values {{Kbn|+|9|2|n}} is a [[Generalized Fermat number]].
    3 KB (432 words) - 18:48, 9 April 2023
  • Woodall prime
    A '''Woodall prime''' is a [[Woodall number]] ({{Kbn|n|2|n}}), which is [[prime]]. A '''Generalized Woodall prime''' could be defined as a Woodall prime with a general base {{Vb}} &gt; 2, so of the form {{Kbn|n|b|n}}.
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  • Cullen prime
    A '''Cullen prime''' is a [[Cullen number]] ({{Kbn|+|n|2|n}}), which is [[prime]]. A '''"Generalized Cullen prime"''' could be defined as a Cullen prime with a general base {{Vb}} &gt; 2, so of the form {{Kbn|+|n|b|n}}.
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  • Cullen prime 2
    {{Cullen prime |CuReserved=PrimeGrid Cullen Prime Search
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  • Williams prime MP least
    ...or any base {{Vb}} with 2 ≤ {{Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b-1)|b|n}}. |category=Williams prime MP without
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  • Williams prime PM least
    ...or any base {{Vb}} with 2 ≤ {{Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|(b+1)|b|n}}. ...ime PM 575|575]] [{{GP|Williams prime PM 575|WiMaxn}}] {{#if:{{GP|Williams prime PM 575|WiReserved}}|<b>RESERVED!</b>}}
    1 KB (161 words) - 13:57, 1 August 2021
  • Williams prime PP least
    ...or any base {{Vb}} with 2 ≤ {{Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b+1)|b|n}}. *{{HistF|2020-07-19|135981 ([[Williams prime PP 327|b=327]])|(unknown),Conjectures 'R Us|551009}}
    2 KB (306 words) - 09:54, 7 September 2020
  • Williams prime MP 362
    {{Williams prime ...s=For all even {{Vn}}-values {{Kbn|+|361|362|n}} is a [[Generalized Fermat number]].
    294 bytes (35 words) - 09:13, 1 August 2021
  • Williams prime MP 10
    {{Williams prime ...]<br>For all even {{Vn}}-values {{Kbn|+|9|10|n}} is a [[Generalized Fermat number]].
    1 KB (131 words) - 07:33, 23 April 2024
  • Williams prime MP 17
    {{Williams prime ...rks=For all even {{Vn}}-values {{Kbn|+|16|17|n}} is a [[Generalized Fermat number]].
    397 bytes (41 words) - 08:25, 1 August 2021
  • Williams prime MM remaining
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    962 bytes (137 words) - 07:42, 1 August 2021
  • Riesel prime small bases least n
    Here are shown the least {{Vn}} &ge; 1 generating a [[Riesel prime]] of the form {{Kbn|k|b|n}} for 2 &le; {{Vb}} &le; 1030 and 2 &le; {{Vk}} & {{HistF|2020-12-14|[[Riesel prime 284 10|{{Kbn|10|284|112809}}]]|Karsten Bonath|566135}}
    6 KB (684 words) - 09:40, 17 March 2024
  • Proth prime small bases least n
    Here are shown the least ''n'' ≥ 1 generating a [[Proth prime]] of the form {{Kbn|+|k|b|n}} for 2 ≤ ''b'' ≤ 1030 and 2 ≤ ''k'' ≤ The data file can be found [[:File:Proth prime small bases least n.csv|here]].
    7 KB (795 words) - 08:03, 5 May 2024
  • Williams prime PP remaining
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | ''n''-value {{#for_external_table:<nowiki/>
    962 bytes (135 words) - 07:49, 26 June 2019
  • Williams prime PM remaining
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    962 bytes (136 words) - 14:00, 1 August 2021
  • Williams prime MP remaining
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    966 bytes (138 words) - 08:03, 1 August 2021
  • Williams prime MP 37
    {{Williams prime ...rks=For all even {{Vn}}-values {{Kbn|+|36|37|n}} is a [[Generalized Fermat number]].
    350 bytes (36 words) - 08:36, 1 August 2021
  • Williams prime MP 50
    {{Williams prime ...rks=For all even {{Vn}}-values {{Kbn|+|49|50|n}} is a [[Generalized Fermat number]].
    292 bytes (31 words) - 08:42, 1 August 2021
  • Proth.exe
    * {{Kbn|+|1|k|n}} ([[Generalized Fermat number]]s) * {{Kbn|±|n|b|n}} ([[Cullen prime|Generalized Cullen]]/[[Woodall prime|Generalized Woodall]] numbers)
    667 bytes (101 words) - 16:44, 31 August 2021
  • Williams prime MP 122
    {{Williams prime ...s=For all even {{Vn}}-values {{Kbn|+|121|122|n}} is a [[Generalized Fermat number]].
    276 bytes (33 words) - 08:48, 1 August 2021
  • Williams prime MP 257
    {{Williams prime ...s=For all even {{Vn}}-values {{Kbn|+|256|257|n}} is a [[Generalized Fermat number]].
    276 bytes (33 words) - 08:52, 1 August 2021
  • Alexander Jones
    ...GIMPS]] with occasional ECM curves and PRP-CF tests, as well as some other prime-searching projects. *[https://github.com/happy5214/rps Riesel Prime Results]
    835 bytes (107 words) - 20:41, 30 April 2024
  • Twin prime least k
    ...bn|+|k|n}} are simultaneously prime, which means they form a set of [[twin prime]]s. The data file can be found [[:File:Twin prime least k.csv|here]].
    993 bytes (142 words) - 09:00, 28 May 2021
  • Paul Leyland
    '''Paul Leyland''' is a British [[number theory|number theorist]] who has studied [[factorization]] and [[primality test]]ing. ...numbers of the form <math>x^y + y^x</math>, which are now called [[Leyland number]]s.
    839 bytes (122 words) - 20:22, 19 July 2019
  • Leyland prime P 4076 1467
    {{Leyland prime {{HistF|2014-07-19|number|Mark Rodenkirch|378541}}
    210 bytes (22 words) - 08:23, 7 August 2019
  • Leyland prime P csv
    Short list of all available [[Leyland number]]s sorted by digits in csv format (Date: {{CURRENTYEAR}}-{{CURRENTMONTH}}-{ |category=Leyland prime P
    468 bytes (64 words) - 08:17, 22 June 2020
  • Leyland number
    A '''Leyland number''' is a number that can be expressed in the form <math>x^y+y^x</math>, where x and y are p A '''Leyland prime''' is a Leyland number which is also a [[prime]] (see {{OEIS|l|A094133}}).
    8 KB (906 words) - 09:59, 5 January 2023
  • Woodall prime P 2 table
    ! No. !! n !! Digits !! Number !! Normalized form !! Discoverer !! Date | 1 || 1 || 1 || {{Kbn|2|1}} || [[Riesel prime 2 1|{{Kbn|2}}]] || n.n. || n.n.
    4 KB (327 words) - 16:31, 31 August 2021
  • Woodall prime M 2 table
    ! No. !! {{Vn}} !! Digits !! Number !! Normalized form !! Discoverer !! Date | 1 || 2 || 1 || {{Kbn|1|2}} || [[Riesel prime 2 1|{{Kbn|2}}]]<ref>Mersenne prime [[M1]]</ref> || n.n. || n.n.
    3 KB (313 words) - 16:51, 31 August 2021
  • RPS Drive 10
    *Name: [[Riesel Prime Search]] Drive 10 ! data-sort-type="number" class="fixhead" | {{Vn}}_min
    2 KB (290 words) - 22:54, 10 April 2024
  • Riesel prime const 514229
    {{Riesel prime const |RcRemarks=514229 is the 29th [[Fibonacci number]].
    371 bytes (30 words) - 09:02, 11 March 2020
  • RPS Megabit Drive 1
    Number of candidates for {{Vn}} &le; 3100000. ! {{Vk}}-value !! [[Nash weight|Nash]] !! Candidates !! {{Vn}}-range || prime(s) for {{Vn}}
    4 KB (439 words) - 10:45, 9 May 2024
  • RPS Megabit Drive 2
    Number of candidates for {{Vn}} &le; 2300000. ! {{Vk}}-value !! [[Nash weight|Nash]] !! Candidates !! {{Vn}}-range || prime(s) for {{Vn}}
    4 KB (376 words) - 21:41, 8 May 2024
  • Riesel prime 2 793
    {{Riesel prime {{HistC|2012-02-21|'''1019935''' not prime|Brian Lody|290227}}, number was proved [https://primes.utm.edu/primes/page.php?id=104867&deleted=1 comp
    1 KB (136 words) - 05:23, 14 March 2023
  • Riesel prime 2 795
    {{Riesel prime {{HistC|2012-02-21|'''1019049''' not prime|Brian Lody|290227}}, number was proved [https://primes.utm.edu/primes/page.php?id=104866&deleted=1 comp
    1 KB (133 words) - 05:30, 14 March 2023
  • PrimeGrid Sierpiński base 5
    Finding primes for the [[Sierpiński number base 5]] problem. |include={Proth prime}:Pk,{Proth prime}:Pk
    2 KB (245 words) - 11:43, 5 September 2021
  • Template prototypes
    ==Template "Riesel prime" [[Riesel new|New]]== *[[:Template:Riesel prime]]{{#dpl:title=Template:Riesel prime|include=#Prototype}}
    1 KB (141 words) - 19:47, 15 March 2024
  • Riesel 2 Count-100
    Riesel numbers {{Kbn|k|n}} with 100 or more prime values {{Vn}}. include={Riesel prime}:Rk,{Riesel prime}:RCount,n100,{Riesel prime}:RNash,maxn
    911 bytes (120 words) - 21:56, 25 July 2021
  • Riesel 2 15k-value
    include={Riesel prime}:Rk,{Riesel prime}:Rk,{Riesel prime}:RCount,{Riesel prime}:RNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 06:59, 16 July 2021
  • Riesel 2 2145k-value
    include={Riesel prime}:Rk,{Riesel prime}:Rk,{Riesel prime}:RCount,{Riesel prime}:RNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max-{{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 07:01, 16 July 2021
  • Riesel 2 2805k-value
    include={Riesel prime}:Rk,{Riesel prime}:Rk,{Riesel prime}:RCount,{Riesel prime}:RNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 07:08, 16 July 2021
  • Riesel 2 Count-0
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with no prime value so far}} Riesel numbers {{Kbn|k|n}} where no prime values are known.
    871 bytes (117 words) - 21:46, 25 March 2024
  • CRUS Liskovets-Gallot
    ...ctures]], which relate to the smallest [[Riesel prime|Riesel]] and [[Proth prime|Proth]] {{Vk}}-values, divisible by 3, with no primes for {{Vn}}-values of ...sted {{Vk}}-values (initially for [[Proth prime]]s, then also for [[Riesel prime]]s), divisible by 3, that had no primes of a given parity. This was proven
    7 KB (957 words) - 22:40, 10 June 2023
  • Riesel 2 3k-value
    include={Riesel prime}:Rk,{Riesel prime}:Rk,{Riesel prime}:RCount,{Riesel prime}:RNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    1 KB (153 words) - 06:55, 16 July 2021
  • Statistics
    Sequences (per base, {{Vk}}-value or individual number) with own page in this Wiki: ...mber || style="text-align:right;"|{{Num|{{#expr:{{PAGESINCATEGORY:Mersenne prime|pages|R}}-2}}}}
    11 KB (1,385 words) - 17:23, 5 April 2024
  • Riesel 2 Missing-range
    ...'' below the largest known prime for that {{Vk}}. This is usually due to a prime being found as part of a project drive, searches for primes with special fo |include={Riesel prime} short2 dpl
    1 KB (192 words) - 09:07, 23 September 2021
  • PrimeGrid Prime Sierpiński Problem
    ...a continuation of the [https://www.mersenneforum.org/showthread.php?t=2665 Prime Sierpinski Project] that operated on the Mersenne Forums. ...7 is the smallest Sierpiński number. However, 78557 itself is not a prime number.
    2 KB (254 words) - 11:43, 5 September 2021
  • Proth 2 3k-value
    include={Proth prime}:Pk,{Proth prime}:Pk,{Proth prime}:PCount,{Proth prime}:PNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 09:18, 23 July 2021
  • Proth prime 3 4
    {{Proth prime ...}}<br>For all even {{Vn}}-values {{Kbn|+|4|3|n}} is a [[Generalized Fermat number]].
    1 KB (120 words) - 21:11, 1 August 2021
  • Riesel problem 2
    The '''2nd Riesel problem''' involves determining the smallest [[Riesel number]]s {{Kbn|k|2|n}} for 509203 &lt; {{Vk}} &lt; 762701, the first and second R [[Category:Riesel prime conjectures|2]]
    4 KB (386 words) - 06:41, 29 March 2024
  • Proth prime 2 121
    {{Proth prime |PRemarks=All values are [[Generalized Fermat number]]s.
    1 KB (112 words) - 19:02, 9 April 2023
  • Proth 2 15k-value
    include={Proth prime}:Pk,{Proth prime}:Pk,{Proth prime}:PCount,{Proth prime}:PNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 09:22, 23 July 2021
  • Proth 2 2145k-value
    include={Proth prime}:Pk,{Proth prime}:Pk,{Proth prime}:PCount,{Proth prime}:PNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max-{{Vn}},\n¦-,,\n¦}
    1 KB (156 words) - 09:36, 23 July 2021
  • Proth 2 2805k-value
    include={Proth prime}:Pk,{Proth prime}:Pk,{Proth prime}:PCount,{Proth prime}:PNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max-{{Vn}},\n¦-,,\n¦}
    1 KB (158 words) - 09:16, 22 March 2024
  • Proth 2 Count-0
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with no prime value so far}} Proth numbers {{Kbn|+|k|n}} where no prime values are known.
    867 bytes (117 words) - 07:46, 26 July 2021
  • Proth 2 Count-100
    Proth numbers {{Kbn|+|k|n}} with 100 or more prime values {{Vn}}. include={Proth prime}:Pk,{Proth prime}:PCount,n100,{Proth prime}:PNash,maxn
    916 bytes (122 words) - 07:51, 26 July 2021
  • Proth 2 Missing-range
    Proth {{Vk}}-values with missing ranges below the largest known prime for that {{Vk}}. |include={Proth prime} short2 dpl
    778 bytes (107 words) - 07:57, 26 July 2021
  • Proth prime 2 25
    {{Proth prime ...arks=For all even {{Vn}}-values {{Kbn|+|25|2|n}} is a [[Generalized Fermat number]].
    2 KB (301 words) - 00:37, 9 March 2023
  • Woodall fill-in
    ...ect by [[Alexander Jones]] for filling-in missing ranges of (Near-)Woodall prime {{Vk}}-values. |include={Riesel prime} short2 dpl
    2 KB (194 words) - 17:44, 11 November 2023
  • PrimeGrid Woodall Prime Search
    The goal of this project is to find [[Woodall number|Woodall prime]]s of the form {{Kbn|n|2|n}}. *[[PrimeGrid Cullen Prime Search]]
    1 KB (150 words) - 11:45, 5 September 2021
  • Proth 2 Low-weight
    include={Proth prime}:Pk,{Proth prime}:PCount,{Proth prime}:PNash,a ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    944 bytes (121 words) - 18:22, 5 August 2021
  • Fermat divisor
    A <b>Fermat divisor</b> is a divisor of a [[Generalized Fermat number]] (in short a "GF Divisor"). ...number]] {{V|F<sub>n</sub>}} is of the form {{Kbn|+|k|n+2}} (so a [[Proth prime]]).
    445 bytes (71 words) - 09:09, 7 September 2021
  • Proth prime 2 289
    {{Proth prime |PRemarks=All primes are also [[Generalized Fermat number#Special conditions for Proth primes|Generalized Fermat primes]].
    1 KB (127 words) - 10:01, 21 September 2021
  • Proth prime 2 49
    {{Proth prime |PRemarks=For all {{Vn}}-values {{Kbn|+|49|2|n}} is a [[Generalized Fermat number]].
    1 KB (144 words) - 22:24, 5 July 2023
  • Amdahl Six
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="UTM prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref> The group has a to
    699 bytes (95 words) - 11:15, 29 November 2021
  • Riesel problem 3
    The '''3rd Riesel problem''' involves determining the smallest [[Riesel number]]s {{Kbn|k|2|n}} for {{Num|762701}} &lt; {{Vk}} &lt; {{Num|777149}}, the se [[Category:Riesel prime conjectures|3]]
    4 KB (337 words) - 18:52, 30 April 2024
  • Even Riesel Problem
    The [[Riesel problem 1|Riesel problem]] is to find the smallest [[Riesel number]] {{Vk}} (odd) such that {{Kbn|k|2|n}} is composite for every {{Vn}} &ge; 1 ...me was found (mostly a higher {{Vn}}). But what about {{Vk}}'s with only a prime with very low {{Vn}}, say {{Vn}} = 1?<br>
    7 KB (718 words) - 10:32, 26 March 2024
  • Proth prime 5 4
    {{Proth prime ...A204322}}. See also {{NWi|MP|5|n}} <br>All primes are [[Generalized Fermat number]]s
    718 bytes (75 words) - 07:17, 14 February 2023
  • Woodall table test
    ! class="fixhead unsortable" | Number at [[Factoring Database|FactorDB]] ! data-sort-type="number" class="fixhead" | Base
    1 KB (162 words) - 14:40, 29 March 2023
  • Twin Prime Search 480000
    Searching for [[twin prime]]s for {{Vn}} values between {{Num|480000}} and {{Num|500000}}. ...s://www.mersenneforum.org/showpost.php?p=624397 2023-02-07]): [[:File:Twin Prime Search primes and twins 480k-500k.txt|LLR result file]]
    2 KB (262 words) - 10:53, 21 March 2024
  • Robert Gerbicz
    ...], [[Riesel prime|Riesel]], and (in an extended form) [[Generalized Fermat number|Generalized Fermat]] primality tests.
    607 bytes (78 words) - 20:45, 8 May 2023
  • PrimeGrid Generalized Fermat Prime Search
    ...t is searching for [[Generalized Fermat number#Dubner|Generalized Fermat]] prime numbers. ...hen this project will jump up to the minimum {{Vb}} needed to surpass that number.<ref>[https://www.primegrid.com/forum_thread.php?id=8422&nowrap=true#125716
    2 KB (310 words) - 06:38, 29 December 2023
  • Twin Prime Search 1700000
    Searching for [[twin prime]]s and [[Sophie Germain prime]]s near {{Vn}} = {{Num|1700000}}. The data file can be found [[:File:Twin Prime Search 1700000.csv|here]] containing 10 primes.
    2 KB (208 words) - 18:36, 30 April 2024
  • Twin Prime Search 3322000
    Searching for [[twin prime]]s and [[Sophie Germain prime]]s near {{Vn}} = {{Num|3322000}}. ...ile can be found [[:File:Twin Prime Search 3322000.csv|here]] containing 1 prime.
    2 KB (204 words) - 18:39, 30 April 2024
  • Riesel problem 4
    The '''4th Riesel problem''' involves determining the smallest [[Riesel number]]s {{Kbn|k|2|n}} for {{Num|777149}} &lt; {{Vk}} &lt; {{Num|790841}}, the th [[Category:Riesel prime conjectures|4]]
    4 KB (336 words) - 18:50, 30 April 2024
  • John Brown
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="T5K prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref>
    410 bytes (48 words) - 23:14, 1 December 2023
  • Bodo Parady
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="T5K prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref>
    413 bytes (48 words) - 23:16, 1 December 2023
  • Gene Smith
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="T5K prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref> He is the brother
    566 bytes (79 words) - 23:21, 1 December 2023
  • Joel Smith
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="T5K prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref> He is the brother
    566 bytes (79 words) - 23:23, 1 December 2023
  • Sergio Zarantonello
    ...a then-record prime number, {{NRi|391581|216193}}, in 1989.<ref name="T5K prime">{{T5000|390|{{Kbn|391581|216193}} on PrimePages}}</ref>
    434 bytes (48 words) - 23:24, 1 December 2023
  • Twin Prime Search 333444
    Searching for [[twin prime]]s and [[Sophie Germain prime]]s near {{Vn}} = {{Num|333444}}. *Number of candidates: {{Num|14464418}}
    2 KB (202 words) - 10:52, 21 March 2024
  • Twin Prime Search 222333
    Searching for [[twin prime]]s and [[Sophie Germain prime]]s near {{Vn}} = {{Num|222333}}. The data file can be found [[:File:Twin Prime Search 222333.csv|here]] containing 3331 primes.
    2 KB (204 words) - 10:51, 21 March 2024
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