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  • 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
  • Splitting a sieve file
    Splitting a sieve file in [[PRP-LLR format]] into a number of separate files using <code>[[wikipedia:AWK|awk]]</code>.
    1 KB (203 words) - 18:52, 2 October 2022
  • Great Internet Mersenne Prime Search
    ...primes are much rarer than ordinary primes, of which there are an infinite number. The GIMPS effort, exhaustively searching for possible candidates since 199
    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 *'''Digits in P<sub>n</sub>''': denotes the [[Perfect number]] 2<sup>n-1</sup> &bull; (2<sup>n</sup>-1) and a downloadable decimal repre
    2 KB (360 words) - 09:44, 6 March 2019
  • Mersenne prime
    More generally, [[Mersenne number]]s (not necessarily primes, but candidates for primes) are numbers that are ...ved that all [[even number|even]] perfect numbers have this form. No [[odd number|odd]] perfect numbers are known, and it is suspected that none exists.
    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
  • Leonhard Euler
    ...his approach to mathematical research becomes apparent. He saw studies of number theory as being vital to the foundations of calculus, and that special func ...to denote the circumference of a circle. Johann Bernoulli represented the number by c. Euler in 1734 denoted it by p, and in a letter of 1736 (in which he f
    16 KB (2,614 words) - 11:48, 14 January 2024
  • Pierre de Fermat
    ...ctures that were later proven or refuted by other mathematicians. [[Fermat number]]s are named after him.
    429 bytes (63 words) - 11:44, 14 January 2024
  • 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
  • Scott Kurowski
    Entropia grew to collaborate with a number of major technology companies including IBM and British Aerospace in the fi
    985 bytes (141 words) - 01:30, 15 January 2024
  • Fermat number
    In [[mathematics]], a '''Fermat number''', named after [[Pierre de Fermat]] who first studied them, is a positive ...up> + 1).) In other words, every prime of the form {{Kbn|+|n}} is a Fermat number, and such primes are called '''Fermat primes'''. The only known Fermat prim
    12 KB (1,913 words) - 14:35, 9 August 2021
  • M50
    | number=467333183359...069762179071 ...mputer network administrator. [https://www.popsci.com/worlds-largest-prime-number-discovered] The primality proof took 6 days of non-stop computing.
    2 KB (333 words) - 13:16, 17 February 2019
  • M49
    | number=300376418084...391086436351 To confirm that there were no errors in the [[hardware]] or [[software]], the number had to be independently verified by running tests on various machines with
    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.
    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.
    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
    [[Category:Number]]
    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 M25 is 2<sup>{{Num|21701}}</sup>-1, a number of {{Num|6533}} [[digit]]s.
    2 KB (303 words) - 11:01, 26 February 2019
  • Lucas-Lehmer test
    ...cas-Lehmer test''' is a deterministic algorithm used to prove a [[Mersenne number]] either composite or prime. It is the last stage in the procedure employed ...<sup>p</sup>-1 would divide into another number, now called a Lucas-Lehmer number denoted S<sub>n</sub> where S<sub>0</sub>=4 and S<sub>n</sub> = (S<sub>n-1<
    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.]] ...a.edu) discovered a new prime [[M47]]. It remained the largest known prime number for almost four and a half years.
    4 KB (564 words) - 00:11, 15 January 2024
  • Digit
    ...g fingers) of the hands correspond to the 10 symbols of the common base 10 number system, i.e. the [[decimal]] (ancient Latin adjective ''dec.'' meaning ten) In a given number system, if the [[base]] is an integer, the number of digits required is always equal to the absolute value of the base.
    1 KB (171 words) - 10:17, 18 January 2019
  • Decimal
    ...base 2. The length of a number (how many [[digit]]s it takes to write the number) depends upon the base.
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    In [[Mathematics]], a '''base''' or '''radix''' is the number of different [[digit]]s that a system of counting uses to represent numbers Bases must be a [[whole number]] bigger than 0. If it was 0, then there would be no digits.
    2 KB (399 words) - 10:37, 18 January 2019
  • Whole number
    The term '''whole number''' does not have a consistent definition. Various authors use it in one of *the positive integers (1, 2, 3, ...) (often called [[natural number]]s)
    413 bytes (54 words) - 09:51, 8 February 2019
  • Integer
    ...r ''Zahlen'' (German for "numbers"). They are also known as the '''[[whole number]]s''', although that term is also used to refer only to the positive intege [[Category:Number systems]]
    3 KB (404 words) - 14:58, 26 March 2023
  • 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
  • Multiplication
    '''Multiplication''' is the process of calculating the result when a number a is [[Addition|added]] to itself b times. The result of a multiplication i ...es are in [[exponent|exponentiation]] (<math>a^0=1</math>) and [[factorial number]]s (0!=1).
    2 KB (271 words) - 17:00, 29 August 2022
  • Exponent
    ...wer a base number is raised to, the exponent is the superscript value of a number written as <math>a^p</math>. ...duct]] a &times; a &times; a &times; a is written as <math>a^4</math>, the number 4 is the index, or exponent.
    1 KB (273 words) - 16:56, 29 August 2022
  • Factorial number
    ...or bang) after a number, it represents multiplying a number by all [[whole number|whole numbers]] smaller than it. *[[Multifactorial number]]
    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
    ...he integers <math>a</math> and <math>b</math> are both greater than 1, the number is composite. *[[Wikipedia:Composite number|Wikipedia]]
    358 bytes (56 words) - 23:30, 26 October 2020
  • Subtraction
    If the minuend is less than the subtrahend, the difference will be a negative number. For example, 17 &minus; 25 = ( &minus;8 ). We can say this as, "Seventeen
    893 bytes (128 words) - 16:58, 29 August 2022
  • Factoring Database
    **Near Cunningham number **Near-repdigit
    1 KB (144 words) - 13:44, 24 January 2019
  • Factorization
    ...ality tests|primality test]], we have to attempt the factorization of this number. The same can be said of the other factor ''b''. So it can be seen that we ..."special purpose" methods whose execution time depends on the size (i.e., number of digits), or on other particular properties of the factors.
    4 KB (642 words) - 12:57, 5 March 2019
  • EFF prizes
    ...substantial award for the person that discovers a ten million digit prime number. ...00}}''' to the first individual or group who discovers a [[Megaprime|prime number with at least '''{{Num|1000000}} decimal digits''']] (awarded 2000-04-06)
    2 KB (321 words) - 18:50, 14 December 2023
  • Number theory
    '''Number theory''' is a branch of pure [[mathematics]] devoted primarily to the stud
    202 bytes (29 words) - 12:55, 20 January 2019
  • 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
    1 KB (198 words) - 07:28, 22 August 2019
  • M48
    | number=581887266232...071724285951 To confirm that there were no errors in the [[hardware]] or [[software]], the number had to be independently verified by running tests on various machines with
    2 KB (235 words) - 11:49, 18 February 2019
  • M47
    | number=316470269330...166697152511 ...er possible usages refer to the [[Nomenclature and notation]] article. The number now refered to as M47 was actually the 45th Mersenne prime found. [[M45]],
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    ...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 ...page4.html Web Archive]). This is slightly shorter than a 24 million digit number. Newer versions of Prime95 (version 25, 26 and 27) can handle Mersenne numb
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Even number
    ...ber of objects can be placed into exactly 2 groups that have the identical number of objects. *[[Odd number]]
    425 bytes (61 words) - 11:19, 7 March 2019
  • Odd number
    An '''odd number''' is any [[integer]] that is not divisible by 2. *[[Even number]]
    316 bytes (42 words) - 11:21, 7 March 2019
  • CUDALucas
    ...UDA]]-based program written by [[Shoichiro Yamada]] for testing [[Mersenne number]]s for primality with [[Lucas-Lehmer test]].
    2 KB (275 words) - 11:11, 21 August 2019
  • Glucas
    ...a program to perform the [[Lucas-Lehmer test]] for primality on [[Mersenne number]]s. It is related to [[GIMPS]] effort to search the largest primes ever fou
    3 KB (426 words) - 14:21, 14 February 2019
  • M41
    | number=299410429404...882733969407 ...senneforum.org/txt/41.txt {{Num|7816230}} decimal digits] long. This prime number was the seventh record prime found by the [[GIMPS]] project.
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    | number=315416475618...411652943871 The number is [http://www.mersenneforum.org/txt/43.txt {{Num|9152052}} decimal digits]
    1 KB (191 words) - 11:31, 18 February 2019
  • M46
    | number=169873516452...765562314751 The number is [http://www.mersenneforum.org/txt/46.txt {{Num|12837064}} decimal digits
    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 ...ng comes into the picture. Using the [[unused computing power]] of a large number of computers some of these problems can actually be solved within a reasona
    4 KB (674 words) - 12:11, 19 February 2019
  • CPU
    *Test to see if one number is larger than another *Move a number from one place to another
    2 KB (366 words) - 09:57, 13 February 2019
  • M42
    | number=122164630061...280577077247 ...senneforum.org/txt/42.txt {{Num|7816230}} decimal digits] long. This prime number was the eighth record prime found by the [[GIMPS]] project.
    934 bytes (118 words) - 11:26, 18 February 2019
  • M1
    | number= 3
    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 ...nd conquer algorithm that recursively breaks down a DFT of any [[Composite number|composite]] size ''n'' = ''n''<sub>1</sub>''n''<sub>2</sub> into many small
    17 KB (2,684 words) - 18:50, 28 September 2023
  • ClLucas
    ...penCL-based program written by [[Shoichiro Yamada]] for testing [[Mersenne number]]s for primality with [[Lucas-Lehmer test]]. It is an OpenCL counterpart of
    1 KB (137 words) - 18:48, 14 December 2023
  • GpuLucas
    ...[[CUDA]]-based program written by [[Andrew Thall]] for testing [[Mersenne number]]s for primality.
    2 KB (239 words) - 11:12, 13 February 2019
  • GpuOwL
    ...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
  • Classes of computers
    ...The architecture of modern supercomputers tend to be built around a large number of "off the shelf" [[CPU]]'s or [[GPU]]'s, rather than those of the past th
    4 KB (558 words) - 22:55, 3 February 2019
  • 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
    The program does not 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 ...index.php?id=1100000001443762221 40,000 digits]. The certification of this number was done by [[Paul Underwood]] with Primo 4.3.0. The certification process
    1 KB (191 words) - 20:33, 12 May 2020
  • M32
    | number=174135906820...328544677887 :2<sup>756 839</sup>-1, a number {{Num|227832}} [[decimal]] [[digit]] long was found to be [[prime]] on 1992
    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 .../sup>-1, which is a number {{Num|378632}} [[decimal]] [[digit]]s long. The number was found to be prime in 1996.
    3 KB (513 words) - 08:42, 18 February 2019
  • Real number
    A '''real number''' is either a [[rational number]] or an [[irrational number]]. The set of real numbers is denoted by <math>\mathbb{R}</math>. *[[Wikipedia:Real_number|Real number]]
    390 bytes (57 words) - 15:00, 26 March 2023
  • Absolute value
    The '''absolute value''' of a [[real number]] is defined as: The absolute value of a [[complex number]] z = x + iy is defined as:
    556 bytes (89 words) - 16:58, 29 August 2022
  • 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
  • Irrational number
    ...an '''irrational number''' is any [[real number]] that is not a [[rational number]], i.e., one that cannot be written as a ratio of two integers, i.e., it is *[[Wikipedia:Irrational_number|Irrational number]]
    763 bytes (124 words) - 15:14, 26 March 2023
  • Binary
    [[Mersenne number]]s when written in binary are all 1's. This makes them [[repunit]] numbers.
    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 | align="center"|[[Mersenne number]]<br>(Base 2 repunit)
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    ...e is no need to exchange data on each step, it is feasible to process each number on a physically separate machine.
    3 KB (416 words) - 06:47, 1 May 2019
  • Processor
    ...in a single chip package. A '''many-core''' processor is one in which the number of cores is large enough that traditional multi-processor techniques are no
    2 KB (269 words) - 14:56, 22 January 2019
  • 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 ...senneforum.org/txt/40.txt {{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 ...senneforum.org/txt/44.txt {{Num|9808358}} decimal digits] long. This prime number was the tenth record prime found by the [[GIMPS]] project.
    997 bytes (129 words) - 11:35, 18 February 2019
  • M45
    | number=202254406890...022308220927 In an interview Hans-Michael Elvenich, a German electrical engineer and prime number enthusiast, stated: "After four years of searching for a prime on [[GIMPS]]
    2 KB (251 words) - 11:40, 18 February 2019
  • Natural number
    *[[Whole number]] *[[Wikipedia:Natural_number|Natural number]]
    316 bytes (43 words) - 15:00, 26 March 2023
  • 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
    *[http://primes.utm.edu/notes/6972593 And the winning number is...]
    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
    ...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
    :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>. There are faster methods, especially when number of thousands or millions of digits are used, as in [[GIMPS]], but they are
    2 KB (339 words) - 18:38, 27 September 2023
  • Modular arithmetic
    We can visualize this arithmetic using a clock. Suppose that the number 12 in the clock is replaced by zero. Then when we have to add an hour, we g where B is the number such that <math>A * B = 1</math> (mod <math>n</math>).
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    The Montgomery representation of a number <math>a</math> is the value ...presentation to normal, just perform a Montgomery multiplication using the number 1 as the second factor.
    4 KB (582 words) - 17:01, 29 August 2022
  • 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 ...er]]. The [[Lucas-Lehmer test]] took 88 hours to run. The primality of the number was confirmed by Slowinski. This showed the effectiveness of [[distributed
    2 KB (224 words) - 11:00, 18 February 2019
  • University of California, Los Angeles
    ...75 years (2 in the very first day of the run, no less). And he raised the number of digits of the largest known [[prime]] (in general) and Mersenne Prime fr
    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''. '' ...ity of California, Los Angeles]]. In 1952, he showed that these [[Mersenne number]]s were all composite except for 17 values of ''n'' = 2, 3, 5, 7, 13, 17,
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    | number=623340076248...743729201151 ...k book. It took Spence's 100 MHz [[Pentium]] computer 15 days to prove the number prime. Alan White Managing Director at Technology Business Solutions, who p
    2 KB (279 words) - 11:01, 18 February 2019
  • Jonathan Pace
    ...umber-discovered How a FedEx employee discovered the world's largest prime number]. ''Popular Science''. 11 Jan 2018. ...-employee.html FedEx employee from Tennessee discovers largest known prime number]. ''CNBC''. 5 Jan 2018.
    2 KB (242 words) - 00:08, 15 January 2024
  • Wacław Sierpiński
    ...h|1882-03-14|1969-10-21}} was a Polish [[mathematician]] contributing in [[number theory]] and others. ...proved there are infinitely many odd integers {{Vk}} (named [[Sierpiński number]]s after him) such that {{Kbn|+|k|n}} is composite for all {{Vn}}.
    592 bytes (86 words) - 00:38, 15 January 2024
  • Sierpiński number
    ...l number {{Vk}} such that all {{Kbn|+|k|n}} for all {{Vn}} are [[Composite number|composite]]. *[[Wikipedia:Sierpiński_number|Sierpiński number]]
    324 bytes (48 words) - 13:37, 8 April 2023
  • 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 ...here ''k'' is one of the remaining 17 (now 5) candidates for [[Sierpiński number]]s smaller than 78557, and ''n'' a positive integer. In order to find such
    3 KB (544 words) - 16:44, 21 July 2019
  • M2
    | number= 7
    193 bytes (19 words) - 13:43, 17 February 2019
  • M3
    | number= 31
    194 bytes (19 words) - 13:43, 17 February 2019
  • M4
    | number= 127
    195 bytes (19 words) - 13:44, 17 February 2019
  • M5
    | number= 8191
    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 ...l factoring]]. Pervushin used the [[Lucas-Lehmer test]] to prove that this number is prime.
    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
    ...umber is exactly divisible. For the L-L test a zero residue means that the number is [[prime]].
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  • GIMPS clients
    So, to test a number efficiently, one must apply the theory to get the tests down to the "weeks" ...came popular among PC enthusiasts and [[Overclocking|overclockers]] as its number-crunching algorithms exercise the computer's processor and memory to their
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  • 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
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  • Mfakto
    ...akt'''oring with '''o'''penCL) is a port of ''[[mfaktc]]'' ('''m'''ersenne number '''fakt'''oring with '''C'''UDA) (for use on ATI/AMD GPUs rather than NVIDI -d <xy> use OpenCL platform number x and device number y
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  • Rho factorization method
    The idea is to create a sequence iterating a polynomial modulo the number to be factored. ...nvented by Richard Brent in 1980 who used it to factor the eighth [[Fermat number]].
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  • Riesel number
    ...value of ''k'' such that {{Kbn|k|n}} is always composite for all [[natural number]]s. In order to demonstrate whether 509203 is the smallest Riesel number or not (the '''[[Riesel problem 1]]'''), a [[distributed computing project]
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  • M12
    | number=170141183460...715884105727 ...d why this happened. Lucas was following a sequence (see [[Double Mersenne number]]). The first possible Mersenne prime (2<sup>1</sup>-1=2), when placed back
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  • Primality test
    ...er a given number is [[prime]] or [[composite number|composite]]. When the number is declared composite, the algorithm does not reveal the prime [[factor]]s. ...(which is far slower than a probable primality test except when the input number has a special form) is run on it.
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  • Lucas primality test
    ...ality test''' invented in 1891 by [[Édouard Lucas]], determines whether a number N is prime or not, using the complete factorization of N-1.
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  • Square root
    ...e non-negative real number whose ''square'' (the result of multiplying the number by itself) is <math>x</math>. ...real numbers, the concept of [[imaginary number|imaginary]] and [[complex number]]s has been developed to provide a mathematical framework to deal with the
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  • Primality testing program
    | [[Mersenne number]]s<br/>a &times; b<sup>n</sup>±c (only factoring and [[probable prime|PRP] | [[generalized Fermat number]]s
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  • 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.
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  • Proof by induction
    ...= F<sub>n-1</sub> + F<sub>n-2</sub> and <math>\phi </math> = 1.61803... a number such that <math>\phi^2 = \phi + 1</math>.
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  • Mprime (Cray)
    ...id Slowinski]] (later versions with [[Paul Gage]]), for testing [[Mersenne number]]s for [[Prime|primality]] on [[Cray Research|Cray]] [[Classes of computers
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  • Probable prime
    ...ecific conditions. While there may be probable primes that are [[Composite number|composite]] (called [[pseudoprime]]s), the condition is generally chosen in ...mality test (like [[Lucas-Lehmer test]]) will be needed to find out if the number is really composite or not.
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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]].
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  • Floating-point
    ...onent]]. The [[base]] for the scaling is normally 2, 10 or 16. The typical number that can be represented exactly is of the form: ...at is, it can be placed anywhere relative to the significant digits of the number. This position is indicated separately in the internal representation, and
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  • Factor5
    ...actoring program|program]] that performs [[Trial factoring]] of [[Mersenne number]]s. It is capable of trial factoring very large numbers, many billions of d :"factor <exponent> <start_bit> <stop_bit> <number of threads to use>"
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  • Complex number
    A '''complex number''' is defined as a pair of [[real number]]s <math>z = (x, y)</math> where the following operations are defined: ...mbers behaves as real numbers. That's why the first element of the complex number is known as the ''real part'' and the second element as the ''imaginary par
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  • Luigi Morelli
    ...upporting [[Trial factoring|factorization]] of large (or small) [[Mersenne number]]s, he wrote the [[Factor5]] [[program]]. As of February 2011, he wrote som ...w [http://www.doublemersennes.org/ website] dealing with [[Double Mersenne number]]s.
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  • Operation Billion Digits
    ...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 ...er the starting one. If you want to do a bigger range, just input a higher number here (be aware that adding a bit depth takes twice the time than the previo
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  • 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]
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  • GIMPS factoring and sieving
    ...a factor than to do the Lucas-Lehmer Test; in fact, over 60% of [[Mersenne number]]s with prime exponents are eliminated from consideration as possible prime ...given Mersenne number up to some predetermined size, usually a prescribed number of bits.
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  • Quadratic residue
    In [[mathematics]], a number {{V|q}} is called a '''quadratic residue''' [[modular arithmetic|modulo]] { In effect, a quadratic residue modulo {{V|p}} is a number that has a [[Modular square root|square root]] in [[modular arithmetic]] wh
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  • 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 ...o zero, there is only one modular square root, namely zero. Otherwise, the number of square roots can be two or zero depending on whether the argument is a [
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  • 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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  • Non-negative
    ...integers from zero upwards, and the non-negative reals are all the [[real number]]s from zero upwards. All whole numbers are non-negative.
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  • Ten million digits
    After the discovery of [[M38]] (the first [[megaprime]] or [[prime]] number greater than 1 million [[decimal]] [[digit]]s) in June of 1999, the next [[ ...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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  • Mmff
    ...man for [[trial factoring]] small [[Fermat number]]s and [[double Mersenne number]]s.
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  • Double Mersenne number
    ...ble Mersenne number''' is a number where the exponent is also a [[Mersenne number]] and usually a [[Mersenne prime]]. These are generally denoted as '''MM<su ...igit]]s long. [[Tony Forbes]] lead an effort to find a [[factor]] for this number. The search has included all '''''[[Value k|k]]''''' values up to {{Num|116
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  • PSearch
    ! scope="col" | Number ...ate, it was the 15th largest prime number, and the 2nd largest Proth prime number.
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  • P-1 factorization method
    Let ''p'' be a prime divisor of the number ''N'' to be factored. If we somehow find a multiple of ''p-1'' we will find ...e method proceeds to compute <math>a^E\,\pmod{N}</math> where ''N'' is the number to factor.
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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. ...instead computes T=S<sup>(6k)<sup>e</sup>-1</sup>, where e is a small even number >2. (6k)<sup>e</sup>-1 = (6k-1)*(6k+1)*(a higher order polynomial in k). Th
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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 ...hand side. A number is a square when all its prime factors appear an even number of times.
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  • Special number field sieve
    {{Shortcut|SNFS|Special number field sieve: special-purpose [[factorization]] algorithm.}} ...(SNFS)''' is a special-purpose [[factorization]] algorithm. The [[general number field sieve]] (GNFS) was derived from it.
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  • Lone Mersenne Hunters
    ...currently being assigned by [[PrimeNet]] in order to eliminate [[Mersenne number]]s as possible [[Mersenne prime]] candidates. This work is suited to older
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  • Fermat pseudoprimality test
    where ''p'' is a [[prime]] number and ''a'' is not multiple of <math>p</math>. ...math>. If the result 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<
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  • Miller-Rabin pseudoprimality test
    ...primality, and <math>N = 2^n\,k + 1</math> where <math>k</math> is an odd number. ...ce is 1 but the previous is not 1 or -1, or the last member is not 1, the number is composite.
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  • Generalized Fermat prime
    A '''generalized Fermat prime''' is a [[generalized Fermat number]] which is [[prime]]. ...s://primes.utm.edu/primes/search.php?Comment=Generalized+Fermat&OnList=yes&Number=10&Style=HTML Current Top10] largest generalized Fermat primes at Prime Pag
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  • Generalized Fermat number
    There are different kinds of '''generalized [[Fermat number]]s'''. :<math>F_{0,r}</math> generates the [[Mersenne number]]s.
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  • M51
    | number=148894445742...325217902591 To confirm that there were no errors in the [[hardware]] or [[software]], the number had to be independently verified by running tests on various machines with
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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
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  • PrimeGrid
    ...ch|Fermat Divisor Search]]: searching for large prime divisors of [[Fermat number]]s. Completed April 2021.
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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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  • FermatSearch
    ...' is a distributed computing project searching for [[factor]]s of [[Fermat number]]s.
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  • Sieve of Eratosthenes
    ...math>. If <math>a \leq \sqrt{N}</math>, then <math>N</math> is [[composite number|composite]]; otherwise it is prime. ...after that will be divisible by <math>2</math>, we cross out every second number; all such numbers are composite.
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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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  • Abundant number
    An '''abundant number''' is any number, '''''n''''', which has a [[sigma|sigma value]] greater than '''''2n'''''. ...bers increase the size of an [[aliquot sequence]] because when an abundant number occurs in a sequence, the next step is larger than the current step. Also,
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  • Prime
    A '''prime number''' (or only '''prime''') is an [[integer]] greater than 1 that is only divi ...r 1. In other words, Q = (2 x 3 x 4 x 5 ... x P) + 1. From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into Q, because
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  • Twin prime
    ...'twin prime''' is a [[prime|prime number]] that differs from another prime number by two, for example the twin prime pair (41, 43). ! Up to !! Number of pairs
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  • Riesel Prime Search
    :# Low-weight {{Vk}}-values that produce a very small number of primes (opposite to (1) above)
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  • Preda Mihăilescu
    His research interests include primality proving ([[GIMPS]]), computational number theory, cryptology, biometry and complex and numerical analysis.
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  • Nomenclature and notation
    ==[[Mersenne number]]== Any number whether [[Composite number|composite]] or [[prime]] of the form <math>2^{x}-1</math>. For one of these
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  • Computing power
    ...of number of [[Processor|cores]], cycles per second each core runs at, and number of [[double-precision]] (64 bit) FLOPS each core can ideally perform. Altho
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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". | Multiline equations <small>(must define number of columns used ({lcr}) (should not be used unless needed)</small>
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  • Seventeen or Bust/Sieving
    Turn proth_sieve on. Input your starting number and the ending one. Just wait after that until it finishes.
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  • Why participate in GIMPS?
    ...nts' approximately 300 BC. His goal was to characterize the even [[perfect number]]s (numbers like 6 and 28 who are equal to the sum of their aliquot divisor Much of elementary number theory was developed while deciding how to handle large numbers, how to cha
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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 [ 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 pre 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
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  • NewPGen
    ...mentations, it is a lot better than performing [[trial factoring]] on each number in the set.
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  • Sophie Germain prime
    A [[prime]] number {{V|p}} is called a '''Sophie Germain prime''' if 2{{V|p}}+1 is also prime.
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  • M13
    | number=686479766013...291115057151
    419 bytes (48 words) - 14:50, 19 September 2021
  • M14
    | number=531137992816...219031728127
    420 bytes (48 words) - 14:49, 19 September 2021
  • M15
    | number=104079321946...703168729087
    430 bytes (49 words) - 14:49, 19 September 2021
  • M16
    | number=147597991521...686697771007
    431 bytes (49 words) - 14:49, 19 September 2021
  • M17
    | number=446087557183...418132836351
    431 bytes (49 words) - 14:49, 19 September 2021
  • M18
    | number=259117086013...362909315071
    415 bytes (47 words) - 22:26, 17 February 2019
  • M11
    | number=162259276829...578010288127
    412 bytes (47 words) - 14:23, 17 February 2019
  • M10
    | number=618970019642...137449562111
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  • Proth prime
    *[[Wikipedia:Proth number|Wikipedia]]
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  • Riesel prime
    *[[Wikipedia:Riesel number|Riesel number]]
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  • Riesel Sieve
    ...oblem]] (which states that {{Vk}}=509203 is the smallest possible [[Riesel number]]). To do that, they have to find primes for all the remaining k values to
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  • M26
    | number=402874115778...523779264511
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  • Floor function
    ...ience, the '''floor function''' is the function that takes as input a real number <math>x</math> and gives as output the greatest [[integer]] less than or eq
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  • Power of two
    A number is said to be a '''power of two''' when its [[factorization]] gives only tw
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  • Introductory stuff
    In mathematics, a '''Mersenne prime''' is a [[prime]] number that is one less than a [[power of two]]. For example, 3 = 4 - 1 = 2<sup>2< ...s of relatively small primes. Trial factoring discovers if there is such a number. If it is, the more expensive Lucas-Lehmer test isn't needed. This type of
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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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  • Overclocking
    ...nd the processor speed. The reason for this, is so one board can operate a number of processors with different speed settings. ...nchmark. The benchmark is fairly diverse and allows the user to change the number of digits of PI that can be calculated from 16 Thousand to 32 Million. The
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  • Triangular number
    ..." (because it is triangular) after a number) represents the summing of a number with all whole numbers smaller than it. *[[Wikipedia:Triangular_number|Triangular number]]
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  • Pocklington's theorem
    ...vented by H. C. Pocklington in 1914, which is a [[primality test]] for the number ''N'', states:
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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> ...hand side. A number is a square when all its prime factors appear an even number of times.
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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
    ...tion system, which makes it easy to use [[PRPclient]] to reserve a testing number directly from [http://prpnet.primegrid.com:12006/ the website]. Then this application will use [[LLR]] or [[PFGW]] to test this testing number and afterwards submit the result back via the website.
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  • 12121 Search
    ...tion system, which makes it easy to use [[PRPclient]] to reserve a testing number directly from [http://prpnet.primegrid.com:12001/ the website]. Then this application will use [[LLR]] or [[PFGW]] to test this testing number and afterwards submit the result back via the website.
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  • Assignments
    where n is the number to reserve. Then, you must set the cache option to reflect the processing t ...as chosen arbitrarily. This is because the reported credit is cut off at a number of significant digits which lies in the range of the actual value for that
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  • GIMPS type of work
    ==Factoring of a prime number (exponent) candidate== ...pically distributed 1-3 curves at a time from PrimeNet (one can change the number of curves by modifying [[Worktodo.txt]]). This is run on smaller exponents
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  • GIMPS networking, PrimeNet and communication
    For nonet computers it is desirable to minimize the number of communication attempts. Set the send new end dates in options / preferen where you substitute the protocol, proxy server domain name and port number exactly as they appear in your web browser. You might need to remove the <c
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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 where k, b, n, c represent a number k &times; b<sup>n</sup> + c. (c can take negative values.)
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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
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  • Bit level
    Since [[Mersenne number]]s are by nature [[binary]], it makes sense to perform calculations on them ...mber binaries are twice as large, while decimals are ten times as large. A number that has 70 binary digits (all 1's) would be at the 70 bit level. To check
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  • Triple check
    ...t match. If the residues match, then the number is known to be [[composite number|composite]], (this assumes that the residue is not zero, otherwise it is [[ ...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
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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
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  • P+1 factorization method
    In order to perform index multiplication by a natural number <math>M</math>, there are two approaches using the formulas shown above. For example for the number 21 which is 10101 when written in binary, you will have the string DDADDA.
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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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  • General number field sieve
    {{Shortcut|GNFS|General number field sieve: most efficient classical [[Factorization|factoring method]] fo ...r field sieve (GNFS)''' is the most efficient classical [[algorithm]] in [[number theory]] for [[Factorization|factoring]] [[integer]]s with 100+ [[digit]]s.
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  • SNFS polynomial selection
    ...wo polynomials with small coefficients and a common root modulo ''N'', the number to be factored. Typically one polynomial is of degree 4, 5 or 6 (the algebr ...lotomic numbers, there are four standard operations you can perform on the number to derive suitable polynomials:
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  • NFSNET
    ...project]] that uses the [[general number field sieve|GNFS]] and [[special number field sieve|SNFS]] [[factorization]] methods to completely factor large num ! Number || Factors
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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
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  • BLS75
    ...orizations of 2^{{V|m}} ± 1"]. ''Mathematics of Computation.'' Volume 29, Number 130: 620-647. The paper shows a number of methods to prove primality of {{V|p}} based on partial factoring of [[P-
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  • M19
    | number=190797007524...815350484991
    280 bytes (23 words) - 22:31, 17 February 2019
  • M20
    | number=285542542228...902608580607
    280 bytes (23 words) - 22:33, 17 February 2019
  • M21
    | number=478220278805...826225754111
    281 bytes (25 words) - 22:35, 17 February 2019
  • M22
    | number=346088282490...883789463551
    281 bytes (25 words) - 22:37, 17 February 2019
  • M23
    | number=281411201369...087696392191
    282 bytes (25 words) - 22:38, 17 February 2019
  • M24
    | number=431542479738...030968041471
    283 bytes (24 words) - 22:48, 17 February 2019
  • M8
    | number=2147483647
    229 bytes (20 words) - 13:50, 17 February 2019
  • M7
    | number=524287
    215 bytes (19 words) - 13:48, 17 February 2019
  • M6
    | number=131071
    215 bytes (19 words) - 13:48, 17 February 2019
  • 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
    7 KB (1,137 words) - 15:30, 13 August 2020
  • Sierpiński-Riesel Base 5
    ...o prove this, one must find primes for all {{Vk}} below the smallest known number.
    2 KB (231 words) - 11:42, 5 October 2020
  • 15k Search
    :Submit your prime number, and list all the programs you've used to find the prime. ...has for the most part, kept the overall lead in the [[The Prime Pages]] by number of primes found. With formidable new competition around, the stakes have be
    3 KB (517 words) - 14:51, 15 February 2019
  • M27
    | number=854509824303...961011228671
    302 bytes (26 words) - 08:16, 18 February 2019
  • M28
    | number=536927995502...709433438207
    278 bytes (23 words) - 08:22, 18 February 2019
  • M29
    | number=521928313341...083465515007
    296 bytes (26 words) - 08:25, 18 February 2019
  • M30
    | number=512740276269...455730061311
    282 bytes (24 words) - 08:28, 18 February 2019
  • M31
    | number=746093103064...103815528447
    285 bytes (24 words) - 08:30, 18 February 2019
  • Milestones report
    ...the status of every [[Mersenne number]] (with a prime exponent) below the number listed is known (it has a known [[factor]], it has been proven prime, or it ::Indicates that every Mersenne number (with a prime exponent), below the number listed, has a known factor or has had at least one Lucas-Lehmer test perfor
    2 KB (298 words) - 12:39, 19 February 2019
  • Mersenne composite
    ...posite''' is any number of the form 2<sup>n</sup>-1 which is a [[composite number]]. ...is always a Mersenne composite, because it is multiple of both [[Mersenne number]]s 2<sup>p</sup>-1 and 2<sup>q</sup>-1.
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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
    7 KB (1,072 words) - 13:10, 19 February 2019
  • Square number
    ...mbers are included, then the ratio of two square integers is also a square number (e.g. 2/3&nbsp;&times;&nbsp;2/3 = 4/9). The number ''m'' is a square number if and only if one can arrange ''m'' points in a square.
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  • Square (algebra)
    In [[algebra]], the square of a number is that number [[multiplication|multiplied]] by itself. It's basically an exponentiation w *[[Square number]]
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  • GIMPS milestones
    | 2001-04-06 || First [[ten million digits]] number sucessfully double-checked. | 2000-09-25 || First [[ten million digits]] number tested for primality.
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  • Sierpiński number base 5
    ...5''' is a value of {{Vk}} such that {{Kbn|+|k|5|n}} is always [[composite number|composite]]. In order to demonstrate whether {{Num|159986}} is the smallest Sierpiński number base 5 or not, a [[distributed computing project]] was created named [[Sier
    556 bytes (83 words) - 10:57, 14 October 2020
  • Riesel number base 5
    ...5''' is a value of ''k'' such that {{Kbn|-|k|5|n}} is always a [[composite number]]. In order to demonstrate whether {{Num|346802}} is the smallest Riesel number base 5 or not, a [[distributed computing project]] was created named [[Sier
    589 bytes (90 words) - 10:30, 26 March 2024
  • 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.
    6 KB (689 words) - 18:14, 4 April 2024
  • Peano postulates
    ...Peano]] in 1889; it characterises the set (class, condition) of [[natural number]]s 1, 2, 3, etc., and consists of the following '''Peano postulates''' (als #''1'' is a natural number
    2 KB (269 words) - 17:04, 24 October 2020
  • 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 ...ch! For the most part we are interested in completing the largest possible number of tests with the available [[CPU]] time, and running one test per [[Proces
    2 KB (392 words) - 14:37, 20 February 2019
  • LL test distribution
    ...7583 GHz-days of [[PrimeNet]] credit upon completion. The largest Mersenne number successfully double checked was 2<sup>{{Num|666666667}}</sup>-1, with {{Num
    2 KB (415 words) - 21:32, 12 February 2020
  • ElevenSmooth
    ...[[distributed computing project]] searching for factors of the [[Mersenne number]] M(3326400) = 2<sup>{{Num|3326400}}</sup>-1.
    366 bytes (42 words) - 12:49, 21 February 2019
  • Mersenne.ca
    ...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 Several graphs are generated nightly, including a graph representing the number of factored vs unfactored exponents and the bit size of the known factors,
    9 KB (1,396 words) - 15:42, 25 February 2019
  • Value k
    When discussing [[Mersenne number]]s, '''all''' [[factor]]s can be expressed in the following form:
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  • Aurifeuillian factor
    When the number is of a particular form (the exact expression varies with the base), Aurife (Number = ''F'' * (''C'' - ''D'') * (''C'' + ''D'') = ''F'' * ''L'' * ''M'')
    10 KB (1,257 words) - 08:04, 24 June 2019
  • 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
  • Factoring program
    *[[Msieve]] (performing Quadratic Sieve and Number Field Sieve) http://www.boo.net/~jasonp/qs.html *[[GGNFS]] (performing Number Field Sieve) http://sourceforge.net/projects/ggnfs
    2 KB (273 words) - 21:53, 8 July 2023
  • Mtsieve
    *mfsieve: search for factors of [[Multifactorial number]]s ...bn|+|k|n}}, remaining terms are potential divisors of [[Generalized Fermat number]]s
    2 KB (338 words) - 06:58, 28 March 2023
  • Number
    [[Category:Number| ]]
    49 bytes (7 words) - 13:26, 6 March 2019
  • Available Double-Triple checks
    ...re listed the available [[worktype|double or triple checks]] of [[Mersenne number]]s.
    703 bytes (67 words) - 21:09, 16 March 2019
  • Carol-Kynea prime
    ...r is a number of the form <math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math>. A Carol/Kynea prime is a [[prime]] wh ! data-sort-type="number" class="fixhead" | Base
    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< *[[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
  • CADO-NFS
    |workload=[[General number field sieve|GNFS]],[[Special number field sieve|SNFS]] '''CADO-NFS''' is a program that implements the number field sieve to factor large integers.
    539 bytes (66 words) - 20:21, 25 July 2020
  • Riesel 2 Low-weight
    ...head"|Count!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    948 bytes (121 words) - 13:08, 21 July 2021
  • Factorial prime
    A '''Factorial prime''' is a [[prime]] of the form '''[[Factorial number]] ± 1'''.
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  • Williams prime MM least
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    2 KB (211 words) - 07:40, 1 August 2021
  • Proth prime 2 1
    |PRemarks=These n-values form the [[Fermat number|Fermat primes]].
    212 bytes (30 words) - 15:35, 2 October 2022
  • Generalized Fermat number 12 5 1
    {{Generalized Fermat number
    128 bytes (12 words) - 09:57, 30 July 2021
  • Generalized Fermat number 12 7 1
    {{Generalized Fermat number
    125 bytes (12 words) - 15:10, 17 August 2021
  • Proth prime 2 9
    ...}}<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]]. *[[Wikipedia:Woodall_number|Woodall number]]
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  • Cullen prime
    A '''Cullen prime''' is a [[Cullen number]] ({{Kbn|+|n|2|n}}), which is [[prime]]. *[[Wikipedia:Cullen_number|Cullen number]]
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  • Cullen prime 2
    18496;23436;C:{{NPr|289|18502}}, [[Generalized Fermat number]]
    2 KB (175 words) - 14:54, 19 September 2021
  • Williams prime MP least
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
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  • Williams prime PM least
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    1 KB (161 words) - 13:57, 1 August 2021
  • Williams prime PP least
    ! data-sort-type="number" class="fixhead" | Base ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
    2 KB (306 words) - 09:54, 7 September 2020
  • Williams prime MP 362
    ...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
    ...]<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
    ...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
    ! data-sort-type="number" class="fixhead" | {{Vk}} ! data-sort-type="number" class="fixhead" | {{Vb}}
    6 KB (684 words) - 09:40, 17 March 2024
  • Proth prime small bases least n
    ...lues given with '-' there is no prime possible or are [[Generalized Fermat number]]s. ! data-sort-type="number" class="fixhead" {{!}} Base
    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
    ...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
    ...rks=For all even {{Vn}}-values {{Kbn|+|49|50|n}} is a [[Generalized Fermat number]].
    292 bytes (31 words) - 08:42, 1 August 2021
  • GMP-Fermat
    ...ag was introduced to gfndsieve to test for Fermat and [[Generalized Fermat number]]s after sieving.
    433 bytes (60 words) - 07:02, 21 June 2019
  • Proth.exe
    * {{Kbn|+|1|k|n}} ([[Generalized Fermat number]]s)
    667 bytes (101 words) - 16:44, 31 August 2021
  • NFS@Home
    ...ey have received enough ECM (and a good enough polynomial is found, if the number is to be factored with GNFS).
    1 KB (211 words) - 18:52, 23 June 2019
  • Williams prime MP 122
    ...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
    ...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
    ...[[PrimeGrid]]. He occasionally factors small composites in the Odd Perfect Number project, and he formerly advanced integer power and other low-difficulty [[
    835 bytes (107 words) - 20:41, 30 April 2024
  • Generalized Fermat number 2 1 116
    {{Generalized Fermat number
    133 bytes (12 words) - 07:54, 18 September 2021
  • Twin prime least k
    ! data-sort-type="number" class="fixhead" | {{Vn}}-value ! data-sort-type="number" class="fixhead" | {{Vk}}-value {{#for_external_table:<nowiki/>
    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
    {{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}}-{
    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
  • Saouter number
    A '''Saouter number''' is a type of [[Generalized Fermat number]]. Numbers of this type have the form ...to this, these numbers share similar properties to those held by [[Fermat number]]s. These numbers were named by [[Tony Reix]]<ref>[https://www.mersenneforu
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  • Mersenne@home
    |result=no primes found, unknown number of factors found
    730 bytes (97 words) - 06:35, 16 August 2019
  • Generalized Fermat number 2 1 107
    {{Generalized Fermat number
    133 bytes (12 words) - 07:49, 18 September 2021
  • Generalized Fermat number 2 1 99
    {{Generalized Fermat number
    127 bytes (12 words) - 19:00, 17 September 2021
  • Generalized Fermat number 2 1 94
    {{Generalized Fermat number
    133 bytes (12 words) - 18:55, 17 September 2021
  • Woodall prime P 2 table
    ! No. !! n !! Digits !! Number !! Normalized form !! Discoverer !! Date
    4 KB (327 words) - 16:31, 31 August 2021
  • James Heinrich
    ...n the PrimeNet server, and coordinates the search of factors of [[Mersenne number]]s with exponents between 1G and 10G (which includes [[Operation Billion Di
    574 bytes (80 words) - 15:00, 21 August 2019
  • Woodall prime M 2 table
    ! No. !! {{Vn}} !! Digits !! Number !! Normalized form !! Discoverer !! Date
    3 KB (313 words) - 16:51, 31 August 2021
  • RPS Drive 10
    ! data-sort-type="number" class="fixhead" | {{Vn}}_min ! data-sort-type="number" class="fixhead" | {{Vn}}_max
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  • Riesel prime const 514229
    |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. *To identify a post of the MersennForum thread the date and post number are given.
    4 KB (439 words) - 10:45, 9 May 2024
  • RPS Megabit Drive 2
    Number of candidates for {{Vn}} &le; 2300000.
    4 KB (376 words) - 21:41, 8 May 2024
  • Riesel prime 2 793
    {{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
    {{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
  • Generalized Fermat number 2 1 147
    {{Generalized Fermat number
    141 bytes (12 words) - 08:56, 18 September 2021
  • Generalized Fermat 262144
    ! data-sort-type="number" class="fixhead" | Number ! data-sort-type="number" class="fixhead" | Digits
    988 bytes (118 words) - 00:24, 6 August 2021
  • PrimeGrid Sierpiński base 5
    Finding primes for the [[Sierpiński number base 5]] problem.
    2 KB (245 words) - 11:43, 5 September 2021
  • Homogeneous Cunningham M 11 5 281
    {{DISPLAYTITLE:Homogeneous Cunningham number 11<sup>281</sup> - 5<sup>281</sup>}} Number: [https://www.mersenneforum.org/showpost.php?p=547092&postcount=42 11<sup>2
    923 bytes (93 words) - 13:34, 15 June 2020
  • Template prototypes
    ==Template "Generalized Fermat number"== ...mplate:Generalized Fermat number]]{{#dpl:title=Template:Generalized Fermat number|include=#Prototype}}
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  • Riesel 2 Count-100
    ...="wikitable sortable"\n!{{Vk}}!!Count!!100th {{Vn}}!!Nash!!data-sort-type="number"|Max {{Vn}},\n¦-,,\n¦}
    911 bytes (120 words) - 21:56, 25 July 2021
  • Riesel 2 15k-value
    ...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
    ...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
    ...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
    ...ead"|{{Vk}}!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    871 bytes (117 words) - 21:46, 25 March 2024
  • Generalized Fermat number 2 1 207
    {{Generalized Fermat number
    124 bytes (12 words) - 12:34, 6 July 2021
  • Generalized Fermat number 3 1 207
    {{Generalized Fermat number
    124 bytes (12 words) - 14:11, 28 July 2021
  • CRUS Liskovets-Gallot
    ! data-sort-type="number" class="fixhead" | Number ! data-sort-type="number" class="fixhead" | {{Vk}}
    7 KB (957 words) - 22:40, 10 June 2023
  • Nash weight
    [[Chris Nash]] gave a weight to show the number of remaining values of {{Kbn|+|k|n}} after sieving the range 100000 < {{Vn}
    2 KB (330 words) - 09:11, 23 September 2021
  • Riesel 2 3k-value
    ...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: | [[:Category:Mersenne prime|Mersenne primes]] || number || style="text-align:right;"|{{Num|{{#expr:{{PAGESINCATEGORY:Mersenne prime
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  • Riesel 2 Missing-range
    ...r>\n!class="fixhead"¦[[Nash weight|Nash]]\n!class="fixhead"¦<abbr title="Number of primes">#</abbr>,,,\n¦}
    1 KB (192 words) - 09:07, 23 September 2021
  • PrimeGrid Prime Sierpiński Problem
    ...7 is the smallest Sierpiński number. However, 78557 itself is not a prime number. ...mber, all prime values of {{Vk}} < 271129 must be shown to produce a prime number of the form {{Kbn|+|k|n}}.
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  • Proth 2 3k-value
    ...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
    ...}}<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
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  • Generalized Fermat number 2 1 38
    {{Generalized Fermat number
    130 bytes (12 words) - 12:32, 6 July 2021
  • Generalized Fermat number 3 2 2
    {{Generalized Fermat number
    122 bytes (12 words) - 15:13, 17 August 2021
  • Generalized Fermat number 2 1 0
    {{Generalized Fermat number
    120 bytes (12 words) - 08:54, 5 July 2021
  • Generalized Fermat number 2 1 1
    {{Generalized Fermat number
    120 bytes (12 words) - 18:11, 1 August 2021
  • Generalized Fermat number 2 1 2
    {{Generalized Fermat number
    122 bytes (12 words) - 08:54, 5 July 2021
  • Generalized Fermat number 2 1 3
    {{Generalized Fermat number
    124 bytes (12 words) - 08:55, 5 July 2021
  • Generalized Fermat number 2 1 4
    {{Generalized Fermat number
    128 bytes (12 words) - 08:55, 5 July 2021
  • Generalized Fermat number 2 1 5
    {{Generalized Fermat number
    140 bytes (12 words) - 00:56, 23 June 2021
  • Generalized Fermat number 2 1 6
    {{Generalized Fermat number
    159 bytes (12 words) - 10:46, 23 June 2021
  • Generalized Fermat number 3 2 3
    {{Generalized Fermat number
    130 bytes (12 words) - 19:57, 1 August 2021
  • Generalized Fermat number 3 1 3
    {{Generalized Fermat number
    129 bytes (12 words) - 14:07, 28 July 2021
  • Generalized Fermat number 2 1 7
    {{Generalized Fermat number
    178 bytes (12 words) - 08:18, 16 September 2021
  • Generalized Fermat number 3 1 2
    {{Generalized Fermat number
    122 bytes (12 words) - 16:02, 17 August 2021
  • Generalized Fermat number 3 1 1
    {{Generalized Fermat number
    120 bytes (12 words) - 15:52, 17 August 2021
  • Generalized Fermat number 3 1 4
    {{Generalized Fermat number
    134 bytes (12 words) - 16:10, 17 August 2021
  • Generalized Fermat number 3 1 5
    {{Generalized Fermat number
    149 bytes (12 words) - 16:14, 17 August 2021
  • Generalized Fermat number 3 1 6
    {{Generalized Fermat number
    169 bytes (12 words) - 16:19, 17 August 2021
  • Generalized Fermat number 2 1 8
    {{Generalized Fermat number
    177 bytes (13 words) - 10:30, 2 July 2021
  • Generalized Fermat number 4 3 2
    {{Generalized Fermat number
    124 bytes (12 words) - 16:32, 17 August 2021
  • Generalized Fermat number 4 3 0
    {{Generalized Fermat number
    120 bytes (12 words) - 16:30, 17 August 2021
  • Generalized Fermat number 3 1 7
    {{Generalized Fermat number
    204 bytes (12 words) - 14:01, 28 July 2021
  • Generalized Fermat number 3 1 8
    {{Generalized Fermat number
    196 bytes (13 words) - 10:05, 16 September 2021
  • Generalized Fermat number 3 1 16408814
    {{Generalized Fermat number
    134 bytes (12 words) - 14:12, 28 July 2021
  • Generalized Fermat number 2 1 9
    {{Generalized Fermat number
    192 bytes (14 words) - 12:21, 2 July 2021
  • Generalized Fermat number 2 1 10
    {{Generalized Fermat number
    206 bytes (14 words) - 11:50, 8 July 2021
  • Generalized Fermat number 2 1 20
    {{Generalized Fermat number
    233 bytes (21 words) - 13:32, 8 July 2021
  • Generalized Fermat number 2 1 24
    {{Generalized Fermat number
    267 bytes (25 words) - 13:34, 8 July 2021
  • Generalized Fermat number 2 1 33
    {{Generalized Fermat number
    117 bytes (12 words) - 14:54, 5 July 2021
  • Generalized Fermat number 2 1 18233954
    {{Generalized Fermat number
    134 bytes (12 words) - 12:35, 6 July 2021
  • Generalized Fermat number 2 1 7963245
    {{Generalized Fermat number
    133 bytes (12 words) - 12:35, 6 July 2021
  • Generalized Fermat number 2 1 103
    {{Generalized Fermat number
    139 bytes (12 words) - 12:33, 6 July 2021
  • Generalized Fermat number 2 1 11075
    {{Generalized Fermat number
    136 bytes (12 words) - 12:34, 6 July 2021
  • Generalized Fermat number 2 1 134995
    {{Generalized Fermat number
    134 bytes (12 words) - 12:34, 6 July 2021
  • Generalized Fermat number 2 1 5523858
    {{Generalized Fermat number
    133 bytes (12 words) - 12:35, 6 July 2021
  • Generalized Fermat number 2 1 11
    {{Generalized Fermat number
    219 bytes (13 words) - 19:36, 6 July 2021
  • Generalized Fermat number 2 1 12
    {{Generalized Fermat number
    288 bytes (20 words) - 11:49, 8 July 2021
  • Generalized Fermat number 2 1 118
    {{Generalized Fermat number
    139 bytes (12 words) - 08:01, 18 September 2021
  • Generalized Fermat number 2 1 28
    {{Generalized Fermat number
    132 bytes (12 words) - 08:08, 7 July 2021
  • Generalized Fermat number 2 1 61
    {{Generalized Fermat number
    129 bytes (12 words) - 08:21, 7 July 2021
  • Generalized Fermat number 2 1 96
    {{Generalized Fermat number
    135 bytes (12 words) - 08:47, 7 July 2021
  • Generalized Fermat number 2 1 142
    {{Generalized Fermat number
    130 bytes (12 words) - 08:35, 18 September 2021
  • Generalized Fermat number 2 1 459
    {{Generalized Fermat number
    136 bytes (12 words) - 09:08, 7 July 2021
  • Generalized Fermat number 2 1 133
    {{Generalized Fermat number
    134 bytes (12 words) - 08:28, 18 September 2021
  • Generalized Fermat number 2 1 635
    {{Generalized Fermat number
    130 bytes (12 words) - 10:41, 7 July 2021
  • Generalized Fermat number 2 1 122
    {{Generalized Fermat number
    130 bytes (12 words) - 08:09, 18 September 2021
  • Generalized Fermat number 2 1 943
    {{Generalized Fermat number
    133 bytes (12 words) - 11:07, 7 July 2021
  • Generalized Fermat number 2 1 27
    {{Generalized Fermat number
    140 bytes (12 words) - 08:18, 8 September 2021
  • Generalized Fermat number 2 1 13
    {{Generalized Fermat number
    279 bytes (19 words) - 07:46, 8 September 2021
  • Generalized Fermat number 2 1 14
    {{Generalized Fermat number
    240 bytes (20 words) - 06:50, 9 July 2021
  • Generalized Fermat number 2 1 15
    {{Generalized Fermat number
    261 bytes (19 words) - 08:40, 16 September 2021
  • Generalized Fermat number 2 1 146
    {{Generalized Fermat number
    131 bytes (12 words) - 08:38, 18 September 2021
  • Generalized Fermat number 2 1 16
    {{Generalized Fermat number
    242 bytes (19 words) - 11:09, 16 September 2021
  • Generalized Fermat number 2 1 17
    {{Generalized Fermat number
    256 bytes (20 words) - 10:26, 9 July 2021
  • Generalized Fermat number 2 1 18
    {{Generalized Fermat number
    238 bytes (19 words) - 11:30, 9 July 2021
  • Generalized Fermat number 11 10 1
    {{Generalized Fermat number
    130 bytes (12 words) - 15:32, 17 August 2021
  • Proth prime 2 121
    |PRemarks=All values are [[Generalized Fermat number]]s.
    1 KB (112 words) - 19:02, 9 April 2023
  • Proth 2 15k-value
    ...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
    ...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
    ...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
    ...ead"|{{Vk}}!!data-sort-type="number" class="fixhead"|Nash!!data-sort-type="number" class="fixhead"|Max {{Vn}},\n¦-,,\n¦}
    867 bytes (117 words) - 07:46, 26 July 2021
  • Proth 2 Count-100
    ...itable sortable"\n!{{Vk}}!!Count!!100th&nbsp;{{Vn}}!!Nash!!data-sort-type="number"|Max&nbsp;{{Vn}},\n¦-,,\n¦}
    916 bytes (122 words) - 07:51, 26 July 2021
  • Proth 2 Missing-range
    ...table"\n!class="fixhead"¦{{Vk}}\n!class="fixhead"¦Date\n!data-sort-type="number" class="fixhead"¦max&nbsp;{{Vn}}\n!class="fixhead"¦max<br>prime\n!class="
    778 bytes (107 words) - 07:57, 26 July 2021
  • Proth prime 2 25
    ...arks=For all even {{Vn}}-values {{Kbn|+|25|2|n}} is a [[Generalized Fermat number]].
    2 KB (301 words) - 00:37, 9 March 2023
  • Generalized Fermat number 11 5 1
    {{Generalized Fermat number
    123 bytes (12 words) - 17:43, 23 August 2021
  • Generalized Fermat number 7 3 2
    {{Generalized Fermat number
    129 bytes (12 words) - 17:48, 23 August 2021
  • Generalized Fermat number 3 1 15
    {{Generalized Fermat number
    216 bytes (13 words) - 08:57, 23 September 2021
  • Generalized Fermat number 3 1 9
    {{Generalized Fermat number
    244 bytes (14 words) - 12:27, 30 July 2021
  • Generalized Fermat number 7 1 1
    {{Generalized Fermat number
    125 bytes (12 words) - 11:48, 22 August 2021
  • Generalized Fermat number 4 3 1
    {{Generalized Fermat number
    125 bytes (12 words) - 23:08, 20 August 2021
  • Generalized Fermat number 3 1 0
    {{Generalized Fermat number
    120 bytes (12 words) - 00:10, 31 July 2021
  • Generalized Fermat number 3 2 1
    {{Generalized Fermat number
    122 bytes (12 words) - 15:19, 17 August 2021
  • Generalized Fermat number 3 2 0
    {{Generalized Fermat number
    120 bytes (12 words) - 15:52, 17 August 2021
  • Generalized Fermat number 7 4 1
    {{Generalized Fermat number
    127 bytes (12 words) - 00:28, 31 July 2021
  • Generalized Fermat number 7 3 0
    {{Generalized Fermat number
    120 bytes (12 words) - 15:52, 17 August 2021
  • Generalized Fermat number 7 6 1
    {{Generalized Fermat number
    127 bytes (12 words) - 00:43, 31 July 2021
  • Generalized Fermat number 8 7 0
    {{Generalized Fermat number
    125 bytes (12 words) - 14:50, 23 August 2021
  • Generalized Fermat number 9 2 1
    {{Generalized Fermat number
    127 bytes (12 words) - 01:03, 31 July 2021
  • Generalized Fermat number 9 7 1
    {{Generalized Fermat number
    127 bytes (12 words) - 01:06, 31 July 2021
  • Generalized Fermat number 9 8 1
    {{Generalized Fermat number
    128 bytes (12 words) - 01:14, 31 July 2021
  • Generalized Fermat number 11 2 1
    {{Generalized Fermat number
    129 bytes (12 words) - 01:18, 31 July 2021
  • Generalized Fermat number 11 3 1
    {{Generalized Fermat number
    128 bytes (12 words) - 01:21, 31 July 2021
  • Generalized Fermat number 11 7 1
    {{Generalized Fermat number
    128 bytes (12 words) - 01:24, 31 July 2021
  • Generalized Fermat number 11 8 1
    {{Generalized Fermat number
    129 bytes (12 words) - 01:31, 31 July 2021
  • Generalized Fermat number 6 1 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:35, 17 August 2021
  • Generalized Fermat number 7 5 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:36, 17 August 2021
  • Generalized Fermat number 12 1 1
    {{Generalized Fermat number
    129 bytes (12 words) - 01:38, 31 July 2021
  • Generalized Fermat number 12 11 1
    {{Generalized Fermat number
    131 bytes (12 words) - 01:43, 31 July 2021
  • Generalized Fermat number 7 2 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:41, 17 August 2021
  • Generalized Fermat number 9 5 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:44, 17 August 2021
  • Generalized Fermat number 11 4 0
    {{Generalized Fermat number
    126 bytes (12 words) - 01:54, 31 July 2021
  • Generalized Fermat number 11 9 0
    {{Generalized Fermat number
    126 bytes (12 words) - 16:54, 17 August 2021
  • Generalized Fermat number 3 2 4
    {{Generalized Fermat number
    136 bytes (12 words) - 19:53, 1 August 2021
  • Generalized Fermat number 3 1 42290
    {{Generalized Fermat number
    128 bytes (12 words) - 20:33, 1 August 2021
  • Generalized Fermat number 3 1 34346
    {{Generalized Fermat number
    128 bytes (12 words) - 20:35, 1 August 2021
  • Generalized Fermat number 2 1
    Factorizations and statistics of [[Fermat number]]s {{V|F}}<sub>{{V|m}}</sub> = {{Kbn|+|1|2|2<sup>m</sup>}} and their factor |category=Generalized Fermat number 2 1 Divs
    2 KB (252 words) - 22:50, 10 September 2021
  • GF Divisors 2 1
    GF Divisors {{Kbn|+|k|2|n}} of [[Fermat number]]s {{V|F}}<sub>{{V|m}}</sub> = {{Kbn|+|1|2|2<sup>m</sup>}}. |category=Generalized Fermat number {{#explode:{{PAGENAME}}| |-2}} {{#explode:{{PAGENAME}}| |-1}} Divs
    595 bytes (83 words) - 22:51, 10 September 2021
  • Woodall fill-in
    ...table"\n!class="fixhead"¦{{Vk}}\n!class="fixhead"¦Date\n!data-sort-type="number" class="fixhead"¦max&nbsp;{{Vn}}\n!class="fixhead"¦max<br>prime\n!class="
    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}}.
    1 KB (150 words) - 11:45, 5 September 2021
  • Proth 2 Low-weight
    ...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
  • Generalized Fermat number 3 1
    Factorizations and statistics of [[Generalized Fermat number]]s {{V|GF}}<sub>(3,1)</sub> = 3<sup>2<sup>n</sup></sup>+1 div 2 and their f |category=Generalized Fermat number 3 1 Divs
    2 KB (261 words) - 22:53, 10 September 2021
  • GF Divisors 3 1
    GF Divisors {{Kbn|+|k|2|n}} of [[Generalized Fermat number]]s {{V|GF}}<sub>(3,1)</sub> = 3<sup>2<sup>n</sup></sup>+1 div 2. |category=Generalized Fermat number {{#explode:{{PAGENAME}}| |-2}} {{#explode:{{PAGENAME}}| |-1}} Divs
    617 bytes (84 words) - 22:54, 10 September 2021
  • Generalized Fermat number 2 1 157167
    {{Generalized Fermat number
    130 bytes (12 words) - 10:22, 16 August 2021
  • Generalized Fermat number 2 1 213319
    {{Generalized Fermat number
    130 bytes (12 words) - 18:54, 16 August 2021
  • Generalized Fermat number 2 1 303088
    {{Generalized Fermat number
    130 bytes (12 words) - 19:29, 16 August 2021
  • Generalized Fermat number 2 1 382447
    {{Generalized Fermat number
    130 bytes (12 words) - 22:01, 16 August 2021
  • Generalized Fermat number 2 1 2145351
    {{Generalized Fermat number
    132 bytes (12 words) - 08:27, 18 August 2021
  • Generalized Fermat number 2 1 2478782
    {{Generalized Fermat number
    132 bytes (12 words) - 09:55, 18 August 2021
  • Generalized Fermat number 5 1 0
    {{Generalized Fermat number
    120 bytes (12 words) - 15:57, 18 August 2021
  • Generalized Fermat number 5 1 1
    {{Generalized Fermat number
    122 bytes (12 words) - 15:59, 18 August 2021
  • Generalized Fermat number 5 1 2
    {{Generalized Fermat number
    124 bytes (12 words) - 16:02, 18 August 2021
  • Generalized Fermat number 5 2 0
    {{Generalized Fermat number
    120 bytes (12 words) - 16:03, 18 August 2021
  • Generalized Fermat number 5 2 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:04, 18 August 2021
  • Generalized Fermat number 5 2 2
    {{Generalized Fermat number
    124 bytes (12 words) - 16:06, 18 August 2021
  • Generalized Fermat number 5 3 0
    {{Generalized Fermat number
    122 bytes (12 words) - 16:13, 18 August 2021
  • Generalized Fermat number 5 3 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:14, 18 August 2021
  • Generalized Fermat number 5 3 2
    {{Generalized Fermat number
    124 bytes (12 words) - 16:16, 18 August 2021
  • Generalized Fermat number 5 3 3
    {{Generalized Fermat number
    130 bytes (12 words) - 16:17, 18 August 2021
  • Generalized Fermat number 5 4 0
    {{Generalized Fermat number
    122 bytes (12 words) - 16:20, 18 August 2021
  • Generalized Fermat number 5 4 1
    {{Generalized Fermat number
    122 bytes (12 words) - 16:21, 18 August 2021
  • Generalized Fermat number 5 4 2
    {{Generalized Fermat number
    124 bytes (12 words) - 16:22, 18 August 2021
  • Generalized Fermat number 2 1 23
    {{Generalized Fermat number
    272 bytes (25 words) - 21:03, 18 August 2021
  • Generalized Fermat number 2 1 36
    {{Generalized Fermat number
    133 bytes (12 words) - 21:40, 18 August 2021
  • Generalized Fermat number 2 1 73
    {{Generalized Fermat number
    122 bytes (12 words) - 17:24, 19 August 2021
  • Generalized Fermat number 2 1 125
    {{Generalized Fermat number
    124 bytes (12 words) - 18:07, 19 August 2021
  • Generalized Fermat number 2 1 1945
    {{Generalized Fermat number
    126 bytes (12 words) - 20:39, 19 August 2021
  • Generalized Fermat number 2 1 3310
    {{Generalized Fermat number
    126 bytes (12 words) - 21:13, 19 August 2021
  • Generalized Fermat number 2 1 23471
    {{Generalized Fermat number
    128 bytes (12 words) - 21:14, 20 August 2021
  • Generalized Fermat number 2 1 125410
    {{Generalized Fermat number
    130 bytes (12 words) - 21:35, 20 August 2021
  • Generalized Fermat number 7 3 1
    {{Generalized Fermat number
    122 bytes (12 words) - 11:09, 22 August 2021
  • Generalized Fermat number 8 5 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:12, 22 August 2021
  • Generalized Fermat number 8 5 1
    {{Generalized Fermat number
    122 bytes (12 words) - 11:13, 22 August 2021
  • Generalized Fermat number 8 5 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:14, 22 August 2021
  • Generalized Fermat number 8 3 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:16, 22 August 2021
  • Generalized Fermat number 8 3 1
    {{Generalized Fermat number
    122 bytes (12 words) - 11:17, 22 August 2021
  • Generalized Fermat number 8 3 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:18, 22 August 2021
  • Generalized Fermat number 7 6 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:30, 22 August 2021
  • Generalized Fermat number 7 6 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:31, 22 August 2021
  • Generalized Fermat number 7 6 4
    {{Generalized Fermat number
    147 bytes (12 words) - 11:33, 22 August 2021
  • Generalized Fermat number 7 5 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:37, 22 August 2021
  • Generalized Fermat number 7 4 0
    {{Generalized Fermat number
    121 bytes (12 words) - 11:38, 22 August 2021
  • Generalized Fermat number 7 4 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:39, 22 August 2021
  • Generalized Fermat number 7 2 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:46, 22 August 2021
  • Generalized Fermat number 7 2 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:47, 22 August 2021
  • Generalized Fermat number 7 1 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:49, 22 August 2021
  • Generalized Fermat number 7 1 2
    {{Generalized Fermat number
    126 bytes (12 words) - 11:50, 22 August 2021
  • Generalized Fermat number 9 8 0
    {{Generalized Fermat number
    122 bytes (12 words) - 11:52, 22 August 2021
  • Generalized Fermat number 9 8 2
    {{Generalized Fermat number
    128 bytes (12 words) - 11:53, 22 August 2021
  • Generalized Fermat number 8 7 1
    {{Generalized Fermat number
    124 bytes (12 words) - 13:04, 22 August 2021
  • Generalized Fermat number 7 1 15
    {{Generalized Fermat number
    173 bytes (13 words) - 14:12, 22 August 2021
  • Generalized Fermat number 4 3 4
    {{Generalized Fermat number
    139 bytes (12 words) - 22:43, 22 August 2021
  • Generalized Fermat number 6 1 0
    {{Generalized Fermat number
    120 bytes (12 words) - 22:44, 22 August 2021
  • Generalized Fermat number 6 1 2
    {{Generalized Fermat number
    126 bytes (12 words) - 22:45, 22 August 2021
  • Generalized Fermat number 6 5 0
    {{Generalized Fermat number
    122 bytes (12 words) - 22:47, 22 August 2021
  • Generalized Fermat number 6 5 1
    {{Generalized Fermat number
    122 bytes (12 words) - 22:47, 22 August 2021
  • Generalized Fermat number 6 5 3
    {{Generalized Fermat number
    132 bytes (12 words) - 22:49, 22 August 2021
  • Generalized Fermat number 6 5 4
    {{Generalized Fermat number
    145 bytes (12 words) - 22:50, 22 August 2021
  • Generalized Fermat number 6 5 2
    {{Generalized Fermat number
    129 bytes (12 words) - 22:55, 22 August 2021
  • Generalized Fermat number 11 5 2
    {{Generalized Fermat number
    130 bytes (12 words) - 23:03, 22 August 2021
  • Generalized Fermat number 7 6 14
    {{Generalized Fermat number
    173 bytes (13 words) - 23:15, 22 August 2021
  • Generalized Fermat number 11 9 15
    {{Generalized Fermat number
    174 bytes (13 words) - 23:20, 22 August 2021
  • Generalized Fermat number 7 1 10
    {{Generalized Fermat number
    169 bytes (13 words) - 23:31, 22 August 2021
  • Generalized Fermat number 8 3 10
    {{Generalized Fermat number
    169 bytes (13 words) - 23:34, 22 August 2021
  • Generalized Fermat number 8 3 16
    {{Generalized Fermat number
    173 bytes (13 words) - 23:39, 22 August 2021
  • Generalized Fermat number 8 5 17
    {{Generalized Fermat number
    175 bytes (13 words) - 23:41, 22 August 2021
  • Generalized Fermat number 9 5 17
    {{Generalized Fermat number
    175 bytes (13 words) - 23:45, 22 August 2021
  • Generalized Fermat number 12 7 16
    {{Generalized Fermat number
    174 bytes (13 words) - 23:48, 22 August 2021
  • Generalized Fermat number 10 3 14
    {{Generalized Fermat number
    179 bytes (13 words) - 00:03, 23 August 2021
  • Generalized Fermat number 11 4 19
    {{Generalized Fermat number
    176 bytes (13 words) - 00:05, 23 August 2021
  • Generalized Fermat number 7 4 21
    {{Generalized Fermat number
    177 bytes (13 words) - 00:12, 23 August 2021
  • Generalized Fermat number 10 9 0
    {{Generalized Fermat number
    123 bytes (12 words) - 06:53, 23 August 2021
  • Generalized Fermat number 10 9 1
    {{Generalized Fermat number
    125 bytes (12 words) - 06:54, 23 August 2021
  • Generalized Fermat number 10 9 2
    {{Generalized Fermat number
    129 bytes (12 words) - 06:55, 23 August 2021
  • Generalized Fermat number 10 7 0
    {{Generalized Fermat number
    123 bytes (12 words) - 06:57, 23 August 2021
  • Generalized Fermat number 10 7 1
    {{Generalized Fermat number
    125 bytes (12 words) - 06:58, 23 August 2021
  • Generalized Fermat number 10 7 2
    {{Generalized Fermat number
    129 bytes (12 words) - 06:59, 23 August 2021

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