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  • Williams prime MM 5
    1 {{HistC|2018-12-06|1 - 321000|Karsten Bonath|501959}}
    917 bytes (86 words) - 12:13, 22 May 2019
  • Williams prime
    ...ber]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1. | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|
    5 KB (744 words) - 07:30, 5 August 2019
  • Carol-Kynea prime 2
    1
    5 KB (537 words) - 08:17, 9 October 2020
  • Carol-Kynea prime 6
    1 1
    1 KB (85 words) - 10:45, 16 April 2023
  • Carol-Kynea prime 10
    1
    1 KB (144 words) - 16:10, 29 March 2024
  • Srsieve
    srsieve -G -n 1 -N 100000 -P 10000000000 "1000*999^n+1" *<code>-n 1</code>: lowest value of ''n'' to search
    2 KB (265 words) - 07:36, 28 May 2021
  • Splitting a sieve file
    BEGIN {getline line; i=1} head[i]=1
    1 KB (203 words) - 18:52, 2 October 2022
  • Examples
    3n+1, & \mbox{if }n\mbox{ is odd} 3n+1, & \mbox{if }n\mbox{ is odd}
    11 KB (1,236 words) - 14:41, 3 September 2020
  • Great Internet Mersenne Prime Search
    ...Mersenne investigated a particular type of prime numbers: 2<sup>p</sup> - 1, in which ''p'' is an ordinary [[prime]].
    3 KB (450 words) - 14:37, 21 August 2019
  • List of known Mersenne primes
    *'''Digits in M<sub>n</sub>''': denotes the [[Mersenne prime]] 2<sup>n</sup>-1 and a downloadable decimal representation ...>''': denotes the [[Perfect number]] 2<sup>n-1</sup> &bull; (2<sup>n</sup>-1) and a downloadable decimal representation
    2 KB (360 words) - 09:44, 6 March 2019
  • Mersenne prime
    ...ime; so is 7 = 8 &minus; 1 = {{Kbn|3}}. On the other hand, 15 = 16 &minus; 1 = {{Kbn|4}}, for example, is not a prime, because 15 is divisible by 3 and :<math>M_n=2^n{-}1</math> .
    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 [[integer]]. ...r <math>2^n{-}1</math> can be calculated by <math>\lfloor{n*log(2)}\rfloor+1</math> (see [[floor function]]).
    2 KB (351 words) - 11:28, 7 March 2019
  • Leonhard Euler
    ...to its diameter in 1755, <math>\large i</math> for the <math>\large\sqrt{-1}</math> in 1777, the notation for finite differences <math>\large\delta y</
    16 KB (2,614 words) - 11:48, 14 January 2024
  • Marin Mersenne
    <math> f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}, </math> ...would be explained if the ratio of the air oscillation frequencies is also 1&nbsp;:&nbsp;2, which in turn is consistent with the source-air-motion-frequ
    11 KB (1,582 words) - 01:17, 15 January 2024
  • Euclid
    ...with the definitions of the most basic and fundamental parts of geometry; 1. A point is that which has no part. 2. A line is breadthless length. From t
    2 KB (341 words) - 11:43, 14 January 2024
  • Laura A. Nickel
    ...with [[Landon Curt Noll]] discovered on 1978-10-30 that 2<sup>21701</sup>-1 was the [[M25|25th Mersenne prime]]. This made international news because N
    2 KB (254 words) - 01:23, 15 January 2024
  • David Slowinski
    | [[M27]] || 2<sup>{{Num|44497}}</sup>-1 || 1979-04-08 | [[M28]] || 2<sup>{{Num|86243}}</sup>-1 || 1982-09-25
    1 KB (213 words) - 23:53, 14 January 2024
  • Fermat number
    :{{V|F}}<sub>{{Vn}}</sub> = {{Kbn|+|1|2|2<sup>n</sup>}} :{{V|F}}<sub>0</sub> = {{Kbn|+|1}} = 3
    12 KB (1,913 words) - 14:35, 9 August 2021
  • M50
    The official discovery date for prime 2<sup>77 232 917</sup>-1 was 2017-12-26. See the [https://www.mersenne.org/primes/press/M77232917.ht *[[Aaron Blosser]] verified it using [[Prime95]] on an Intel Xeon server in 1.5 days
    2 KB (333 words) - 13:16, 17 February 2019
  • M49
    The official discovery date for 2<sup>{{Num|74207281}}</sup>-1 was 2016-01-07. See the [http://www.mersenne.org/primes/?press=M74207281 pr
    2 KB (283 words) - 11:50, 18 February 2019
  • Curtis Cooper
    ...prime]], 2<sup>32 582 657</sup>-1. As of 2008-09-15 his account is ranked #1 on [[PrimeNet]] in [[Lucas-Lehmer test|LL testing]], with over 242 000 P90 ...3-01-25 Cooper discovered his third Mersenne prime, 2<sup>57 885 161</sup>-1, the [[M48|48th]] known.
    2 KB (237 words) - 11:34, 14 January 2024
  • Derrick Henry Lehmer
    ...vement to the [[Lucas primality test]] for [[Mersenne prime]]s <math>2^p{-}1</math>, extending its application to all odd prime exponents ''p'', and ena
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Base 2 Home Prime Results
    {| border="1" cellpadding="4px" style="border:3px; border-color:#000; border-collapse:co {| border="1" cellpadding="4px" style="border:3px; border-color:#000; border-collapse:co
    2 KB (175 words) - 18:45, 14 December 2023
  • Cunningham project
    ...to find the complete [[factorization]] of numbers of the form <math>b^n\pm 1</math> for <math>b</math> = 2, 3, 5, 6, 7, 10, 11, 12. The values of the ex :<math>(b^{kn}-1) = (b^n-1) \sum _{r=0}^{k-1} b^{rn}</math>
    7 KB (1,150 words) - 23:48, 19 April 2023
  • 2 Minus Tables
    ==Factorizations Of Cunningham Numbers C<sup>-</sup>(2,n) = 2<sup>n</sup> - 1== * 001 - 100 : {{FDBCunningham|2|-|1|100}}
    2 KB (176 words) - 12:01, 13 February 2019
  • M25
    M25 is 2<sup>{{Num|21701}}</sup>-1, a number of {{Num|6533}} [[digit]]s. ...heory and that Tuckerman's discovery of [[M24]] (2<sup>{{Num|19937}}</sup>-1) was the start of this island.
    2 KB (303 words) - 11:01, 26 February 2019
  • Lucas-Lehmer test
    ...905 - 1991) provided a complete proof that this was not only true when p = 1 (mod 4), but for all odd prime exponents. The test therefore takes its name ...> - 1 divides S<sub>3</sub> (37634 / 31 = 1214) shows that 2<sup>5</sup> - 1 is prime.
    20 KB (3,572 words) - 14:30, 17 February 2019
  • Édouard Lucas
    ...ucas-Lehmer test]]. In 1876, Lucas proved the primality of <math>2^{127}{-}1</math> ([[M12]]) and this remained the highest [[Mersenne prime]] for almos
    2 KB (296 words) - 01:09, 15 January 2024
  • Decimal
    ...rious symbols (called [[digit]]s) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large.
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    | 1 || '''1''' || 1
    2 KB (399 words) - 10:37, 18 January 2019
  • Whole number
    *the nonnegative [[integer]]s (0, 1, 2, 3, ...) *the positive integers (1, 2, 3, ...) (often called [[natural number]]s)
    413 bytes (54 words) - 09:51, 8 February 2019
  • Integer
    ...positive [[natural number]]s (1, 2, 3, &hellip;), their negatives (&minus;1, &minus;2, &minus;3, ...) and the number zero. The set of all integers is u ...e operation of [[division]], since the quotient of two integers (''e.g.'', 1 divided by 2), need not be an integer.
    3 KB (404 words) - 14:58, 26 March 2023
  • Addition
    :1 + 5 = 6
    333 bytes (43 words) - 16:55, 29 August 2022
  • Multiplication
    ...ponent|exponentiation]] (<math>a^0=1</math>) and [[factorial number]]s (0!=1).
    2 KB (271 words) - 17:00, 29 August 2022
  • Exponent
    ...<math>a^p</math> means that we are notating the number <math>\large \frac{1}{a*a*a*a...}</math> where, you guess it, the [[absolute value]] of p repres ...th> equals the reciprocal (or the multiplicative inverse) of a, that means 1/a.
    1 KB (273 words) - 16:56, 29 August 2022
  • Factorial number
    :<math>n! = 1 \cdot 2 \cdot 3 \cdots (n{-}2) \cdot (n{-}1) \cdot n</math> for <math>n \ge 1</math>.
    729 bytes (93 words) - 13:40, 5 November 2023
  • Factor
    ...is a [[prime]] number, and 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
    ...ive [[integer]] is '''composite''' if it is neither [[prime]] nor equal to 1. The smallest composite is 4. ...where the integers <math>a</math> and <math>b</math> are both greater than 1, the number is composite.
    358 bytes (56 words) - 23:30, 26 October 2020
  • Factoring Database
    **Greatest prime factor ^2+1, ^2+2, ^2-1, ^2-2, ^3+1, ^3-1
    1 KB (144 words) - 13:44, 24 January 2019
  • Factorization
    ...ion method|p-1]]: It finds a factor ''p'' if the largest prime factor of p-1 is small. *[[p+1 factorization method|p+1]]: Similar to p-1, but succeeds if p+1 has no large factors.
    4 KB (642 words) - 12:57, 5 March 2019
  • M48
    The official discovery date for 2<sup>{{Num|57885161}}</sup>-1 was 2013-01-25. See the [http://www.mersenne.org/primes/?press=M57885161 pr .../watch?v=QSEKzFGpCQs New Largest Known Prime Number 2<sup>57,885,161</sup>-1] at YouTube channel Numberphile
    2 KB (235 words) - 11:49, 18 February 2019
  • M47
    ...ormally refers to the 47th [[Mersenne prime]] 2<sup>{{Num|43112609}}</sup>-1, in order of size from the smallest to greatest. This is the primary usage On 2018-04-08 all tests below 2<sup>{{Num|43112609}}</sup>-1 were verified by [[GIMPS]], officially making it the 47th Mersenne prime.
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    ...ehmer test|LL]], PRP, [[Trial factoring|TF]], [[P-1 factorization method|P-1]], [[Elliptic curve method|ECM]]|release=1996|latest=30.3b6<br/><small>2020 {| style="font-size: 85%; text-align: center" border="1" style="border: 1px solid #afafaf; background-color: #f9f9f9; border-collap
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Odd number
    When expressed in decimal notation, the odd numbers end in 1, 3, 5, 7 or 9. All prime numbers except 2 are odd.
    316 bytes (42 words) - 11:21, 7 March 2019
  • CUDALucas
    CUDALucas -cufftbench 1 22680 5 CUDALucas -threadbench 1 22680 5 10
    2 KB (275 words) - 11:11, 21 August 2019
  • M41
    ...used to refer to the 41st [[Mersenne prime]] 2<sup>{{Num|24036583}}</sup>-1. ...nce using half of a Bull NovaScale 5000 HPC running Linux on 16 Itanium II 1.3 GHz CPUs for five days using the [[Glucas]] program by Guillermo Balleste
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    ...used to refer to the 43rd [[Mersenne prime]] 2<sup>{{Num|30402457}}</sup>-1. *by Tony Reix of Bull S.A. in Grenoble, France, in 5 days using 16 Itanium2 1.5 GHz [[CPU]]s of a Bull NovaScale 6160 HPC at Bull Grenoble Research Cente
    1 KB (191 words) - 11:31, 18 February 2019
  • M46
    ...used to refer to the 46th [[Mersenne prime]] 2<sup>{{Num|42643801}}</sup>-1.
    2 KB (248 words) - 11:45, 18 February 2019
  • Riesel prime 2 102765
    1
    935 bytes (70 words) - 18:56, 10 December 2022
  • Riesel prime 2 187605
    1;T:S
    490 bytes (35 words) - 12:22, 11 December 2022
  • CPU
    ...uction takes a very small time to happen. Many CPUs today can do more than 1 billion instructions in a single second. In general, the more a CPU can do
    2 KB (366 words) - 09:57, 13 February 2019
  • Program
    *Knuth, Donald E., The Art of Computer Programming, Volume 1, 3rd Edition, 1997, Addison-Wesley, ISBN 0-201-89683-4
    2 KB (263 words) - 11:53, 7 February 2019
  • GPU
    :P-1 testing
    2 KB (250 words) - 08:44, 13 February 2019
  • M42
    '''M42''' refers to the 42nd [[Mersenne prime]] 2<sup>{{Num|25964951}}</sup>-1.
    934 bytes (118 words) - 11:26, 18 February 2019
  • M1
    | rank= 1 | digits= 1
    193 bytes (19 words) - 13:43, 17 February 2019
  • Conjectures 'R Us
    ...|Riesel value]]' (-1 form) that is composite for all values of {{Vn}} &ge; 1. Conjectures must have a finite covering set. {{Vk}}-values are not conside ==Sub-project #1==
    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 ...f_j = \sum_{k=0}^{n-1} x_k e^{-{2\pi i \over n} jk } \qquad j = 0, ... ,n-1.</math>
    17 KB (2,684 words) - 18:50, 28 September 2023
  • GpuOwL
    ...://github.com/preda/gpuowl/tree/V1 gpuOwL V.1.x branch] at GitHub (version 1 uses 4M FFT and is about 50% faster than version 2) [http://www.mersennefor
    1 KB (216 words) - 05:22, 1 December 2020
  • M32
    :2<sup>756 839</sup>-1, a number {{Num|227832}} [[decimal]] [[digit]] long was found to be [[prime
    2 KB (279 words) - 08:35, 18 February 2019
  • M33
    '''M33''' refers to 33rd [[Mersenne prime]] number 2<sup>{{Num|859433}}</sup>-1.
    814 bytes (97 words) - 08:38, 18 February 2019
  • M34
    ...and in order of discovery. Specifically M34 is 2<sup>{{Num|1257787}}</sup>-1, which is a number {{Num|378632}} [[decimal]] [[digit]]s long. The number w
    3 KB (513 words) - 08:42, 18 February 2019
  • 2 Plus Tables
    ==Factorizations Of Cunningham Numbers C<sup>+</sup>(2,n) = 2<sup>n</sup> + 1== * 001 - 100 : {{FDBCunningham|2|+|1|100}}
    2 KB (127 words) - 15:28, 17 August 2019
  • Rational number
    :<math>\large a + \frac{k(b-a)}{n+1}</math> by varying the number <math>k</math> from 1 to <math>n</math>. Then we can make the value <math>n</math> as high as we
    3 KB (541 words) - 15:01, 26 March 2023
  • Binary
    ...em, a representation for numbers using only two [[digit]]s (usually, 0 and 1). Thus it is a [[base]] 2 numbering system. ...the next digit to the right; the place value of the rightmost digit being 1.
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...digit]]. All [[Mersenne number]]s are repunit ('''rep'''eated '''unit''', "1" being the number referred to as "unity") numbers. 111 is a repunit, in bas :(10<sup>n</sup> - 1) / 9
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    ==Example 1== ! Step !! Input 1 !! Operation !! Input 2 !! Result !! 1440<br>x 365
    3 KB (416 words) - 06:47, 1 May 2019
  • Trial factoring
    ...last prime factor possibility for some number N would be P(m) where P(m + 1) squared exceeds N. ...factor candidates would be close to <math>\frac {\sqrt{N}}{Ln(\sqrt{N}) - 1}</math> which for <math>N = 10^{20}</math> is 450 million.
    7 KB (1,221 words) - 13:20, 11 February 2019
  • M40
    ...e 40th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|20996011}}</sup>-1. This number was discovered to be [[prime]] on 2003-11-17 by [[Michael Shaf ..., California (author of program [[Mlucas]]) using three weeks of time on a 1 GHz HP Alpha workstation.
    1 KB (189 words) - 11:17, 18 February 2019
  • Michael Shafer
    ...scovered the [[M40|40th]] [[Mersenne prime]], 2<sup>{{Num|20996011}}</sup>-1 at [[GIMPS]] project.
    660 bytes (88 words) - 00:39, 15 January 2024
  • Josh Findley
    ...very of the [[M41|41st known Mersenne prime]] 2<sup>{{Num|24036583}}</sup>-1.
    695 bytes (93 words) - 11:46, 14 January 2024
  • M39
    | top5000id=1 ...e 39th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|13466917}}</sup>-1. This number was discovered to be [[prime]] on 2001-11-14 by [[Michael Came
    868 bytes (109 words) - 11:14, 18 February 2019
  • M44
    ...rime]]. Currently that designation belongs to 2<sup>{{Num|32582657}}</sup>-1.
    997 bytes (129 words) - 11:35, 18 February 2019
  • M45
    '''M45''' normally refers to 2<sup>{{Num|37156667}}</sup>-1, the 45th [[Mersenne prime]] in order of size from the smallest to greatest
    2 KB (251 words) - 11:40, 18 February 2019
  • Odd Magnar Strindmo
    ...46th Mersenne prime]] (chronologically 47th), 2<sup>{{Num|42643801}}</sup>-1. Strindmo goes by the alias '''Stig M. Valstad''' on [[GIMPS]].
    991 bytes (141 words) - 00:33, 15 January 2024
  • M38
    ...he 38th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|6972593}}</sup>-1. This number was discovered to be [[prime]] on 1999-06-01 by [[Nayan Hajrat
    1 KB (165 words) - 11:10, 18 February 2019
  • Nayan Hajratwala
    ...ho discovered the [[M38|38th Mersenne prime]], 2<sup>{{Num|6972593}}</sup>-1.
    809 bytes (109 words) - 23:55, 14 January 2024
  • Coprime
    ...ly primes when their [[greatest common divisor]] is 1 (<math>\gcd{(x,y)} = 1</math>). This does not mean that any of these numbers is prime.
    738 bytes (112 words) - 09:50, 23 January 2019
  • Greatest common divisor
    When the greatest common divisor is 1, both numbers are [[coprime]] or relatively prime. This does not mean that #Go back to step 1.
    2 KB (339 words) - 18:38, 27 September 2023
  • Modular arithmetic
    ...can be done when working modulo N, where N is an [[integer]] greater than 1. ...s is arithmetic modulo 12 and the set of numbers representing the hours 0, 1, 2, 3,..., 11 is known as <b>Z</b>/12<b>Z</b>.
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    ...onentiation]], [[Elliptic curve method|ECM]], [[P-1 factorization method|p-1]], etc.) this method is really fast. ...ation 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
    :<math>O(\exp{\sqrt{(\log p \,\log \log p)(1+O(1)}})</math> ...omposite number is a number that has divisors that are neither itself, nor 1. A highly composite number is a number that has lots and lots of divisors.
    19 KB (3,181 words) - 22:27, 6 July 2023
  • M35
    Specifically 2<sup>{{Num|1398269}}</sup>-1, written out in full [http://www.mersenneforum.org/txt/35.txt {{Num|420921}
    2 KB (224 words) - 11:00, 18 February 2019
  • Raphael M. Robinson
    ...t|Lucas-Lehmer]] [[primality test]] to determine whether 2<sup>''n''</sup>-1 was prime for all prime ''n'' < 2304 on a [[SWAC (computer)|SWAC]] at [[Uni ....htm In memoriam : Raphael Mitchel Robinson,]" ''Bull. Symbolic Logic'' '''1''': 340-43.
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    ...he 36th [[Mersenne prime]], specifically it is 2<sup>{{Num|2976221}}</sup>-1. This number was dicovered to be [[prime]] on 1997-08-24 by [[Gordon Spence ...umber]] is 2<sup>{{Num|2976220}}</sup> &bull; (2<sup>{{Num|2976221}}</sup>-1). This number is {{Num|1791864}} digits long.
    2 KB (279 words) - 11:01, 18 February 2019
  • Sierpiński problem
    *{{Kbn|+|78557|4n+1}} is multiple of 5. *{{Kbn|+|78557|3n+1}} is multiple of 7.
    5 KB (650 words) - 10:25, 26 March 2024
  • Paul Seelhoff
    ...50?tify={%22pages%22:%5B306%5D,%22view%22:%22%22} "Generalregister zu Band 1-50 der Zeitschrift für Mathematik und Physik"], p.292) ...fy={%22pages%22:%5B412%5D,%22view%22:%22%22} "Die Zahlen von der Form k.2n+1"], Zeitschrift fur Mathematik und Physik, '''Vol. 31''' (1886) p380
    2 KB (195 words) - 00:13, 15 January 2024
  • Covering set
    *[[Riesel problem 1|Riesel problem]]
    380 bytes (56 words) - 10:27, 26 March 2024
  • Seventeen or Bust
    |result=11 k's eliminated as a standalone project, 1 k eliminated as a subproject on PrimeGrid The aim of the project is to find [[prime]]s of the form <math>k*2^n+1</math>, where ''k'' is one of the remaining 17 (now 5) candidates for [[Sie
    3 KB (544 words) - 16:44, 21 July 2019
  • M2
    | digits= 1
    193 bytes (19 words) - 13:43, 17 February 2019
  • Perfect number
    ...roper 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. ...numbers are generated by the formula 2<sup>''n''-1</sup>(2<sup>''n''</sup>-1):
    6 KB (885 words) - 11:33, 7 March 2019
  • M9
    The ninth [[Mersenne prime]], 2<sup>61</sup>-1 or {{Num|2305843009213693951}}. ...mber, ([[Édouard Lucas]] having shown earlier that [[M12]], <math>2^{127}-1</math> is also prime), and it remained so until 1911. Prior to the develope
    2 KB (213 words) - 14:30, 17 February 2019
  • Residue
    ...of the L-L test. When the numbers being tested are large: <math>\gt2^{64}-1</math> (i.e. exponents larger than 64) and above, the residue is 16 hexadec Here is the Lucas test for <math>2^7-1</math>, which is 127:
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    ! LL test !! PRP test !! Trial factoring !! ECM factoring !! P-1 factoring ...mount of factoring work using (say) Pollard's [[p-1 factorization method|p-1 method]] or the [[elliptic curve method]], both of which involve manipulati
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...CPU resources. However GPU sieving is not supported on compute capability 1.x GPUs in 0.20. Mfaktc 0.21 can do GPU sieving for those old GPUs. *A [[CUDA]] capable GPU with compute capability 1.1 (or newer), any Geforce 8000, 9000, 200, 400, 500 series ''except'' those w
    5 KB (765 words) - 14:54, 25 February 2019
  • Mfakto
    -v <n> verbosity level: 0=terse, 1=normal, 2=verbose, 3=debug -st run built-in selftest (about 1,500 testcases) and exit
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    *x<sub>1</sub> = f(x<sub>0</sub>) *x<sub>2</sub> = f(x<sub>1</sub>)
    3 KB (558 words) - 10:28, 6 February 2019
  • Riesel number
    ...ether 509203 is the smallest Riesel number or not (the '''[[Riesel problem 1]]'''), a [[distributed computing project]] was created named [[Riesel Sieve *[[Riesel problem 1]]
    827 bytes (112 words) - 08:21, 25 March 2024
  • M12
    ...t [[University of California, Los Angeles]] found [[M13]], 2<sup>521</sup>-1. .../sup>-1=7) also produces a prime. When this value is tested (2<sup>7</sup>-1=127), another prime is produced. So, Lucas was testing to see if this trend
    2 KB (354 words) - 14:52, 19 September 2021
  • Primality test
    *[[Lucas primality test|Lucas Test]]: Used when the number {{V|N}}-1 is completely factored. ...factors of the input number - 1 are known (the unfactored part of {{V|N}}-1 must be less than the [[square root]] of {{V|N}}).
    3 KB (501 words) - 05:20, 3 August 2021
  • Lucas primality test
    ...whether a number N is prime or not, using the complete factorization of N-1. ...>(N-1)/q</sup> is not congruent to 1 modulo N for any prime divisor q of N-1, then N is a prime.
    1 KB (177 words) - 14:31, 17 February 2019
  • Square root
    ...gth of the [[diagonal]] of a [[square (geometry)|square]] with side length 1. ...s to the equation <math>x = \sqrt{x}</math> The solution set is <math>\{0, 1\}</math>.
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Pépin's test
    ...s, like the [[Generalized Fermat number]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1</math> with k = 5 instead of k = 3. ...^{2^n}+1</math> is a prime if and only if <math>\ 3^{(F_n-1)/2} \ \equiv -1 \ \pmod{F_n}</math>.
    2 KB (401 words) - 14:40, 6 March 2019
  • Proof by induction
    *'''Step 1''' ...of n, say n<sub>0</sub>. This value n<sub>0</sub> is normally taken to be 1, but that is not essential. In some proofs (see example 4 below) we have to
    4 KB (679 words) - 13:57, 20 February 2019
  • Probable prime
    .... If the result is different from 1, <math>n</math> is composite. If it is 1, <math>n</math> may or may not be prime; <math>n</math> is then called a (w
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  • Pseudoprime
    ...ns of the Miller-Rabin test, for example, has a probability of only <math>{1/4}^{100}</math> of being composite, which is less than <math>10^{-60}</math
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  • Complex number
    ...> we get <math>(-1, 0)</math>. Since no real number is the square root of -1, we can now understand why the second element is the imaginary part. ...<math>z = x + iy</math>. From the previous paragraph we get: <math>i^2 = -1</math>.
    2 KB (280 words) - 14:59, 26 March 2023
  • Operation Billion Digits
    ...al project; [[p-1 factorization method|p-1]], [[p+1 factorization method|p+1]], and [[Elliptic curve method|ECM]] tests are also unfeasible because they ...nd output its progress by k, d and bit depth. The k is, well, the k on 2kp+1, where p is the exponent you're searching. The d is the divisor tried and t
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  • M37
    ...he 37th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|3021377}}</sup>-1. This number was discovered to be [[prime]] on 1988-01-27 by [[Roland Clark
    877 bytes (111 words) - 11:04, 18 February 2019
  • Team Prime Rib
    ...0 CPU years of [[GIMPS factoring and sieving#Trial_factoring|factoring]] (#1 on [[PrimeNet]]).
    620 bytes (88 words) - 11:46, 12 February 2019
  • GIMPS factoring and sieving
    ...re capable of doing three different types of factoring: trial factoring, p-1 and ECM, but only the first two of these types are routinely done on GIMPS ...teger <math>k</math> and furthermore must also leave a remainder of either 1 or 7 upon division by 8. See below for an explanation.
    6 KB (962 words) - 10:08, 7 March 2019
  • Modular square root
    ...eger]] number <math>a</math> modulo an integer <math>m</math> greater than 1 is an integer such that: ...s odd, we assume that the quantity <math>a^{(m-1)/2} \bmod m</math> equals 1 (otherwise there is no square root if <math>a \not\equiv 0\ \pmod m</math>)
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  • Legendre symbol
    *1 if <math>a</math> is a square [[modular arithmetic|modulo]] <math>p</math> *&minus;1 if <math>a</math> is not a square modulo <math>p</math>, or in other words
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  • Law of quadratic reciprocity
    *If at least one of <math>p</math> or <math>q</math> are congruent to 1 mod 4: <math>p</math> is a quadratic residue modulo <math>q</math> if and o This does not cover the cases where we want to know whether -1 or 2 are quadratic residues or non-residues modulo <math>p</math>.
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  • Ten million digits
    ...overy of [[M38]] (the first [[megaprime]] or [[prime]] number greater than 1 million [[decimal]] [[digit]]s) in June of 1999, the next [[EFF prizes]] fo
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  • LLR
    **{{V|N}}-1 [[Pocklington algorithm]] for {{Kbn|+|k|b|n}} numbers. **{{V|N}}+1 [[Morrison algorithm]] for {{Kbn|k|b|n}} numbers.
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  • Double Mersenne number
    ...'MM<sub>p</sub>''', '''MMp''', or '''MM(p)''' and refer to <math>2^{2^p-1}-1</math>. Early on it was thought that if M(p) was prime so too was MM(p). *MM(2) = <math>2^3-1</math> = 7, known prime since antiquity
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  • CUDAPm1
    ...ram, based on [[CUDALucas]] code, for testing [[P-1 factorization method|P-1]] testing on [[GPU]]s.
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  • P-1 factorization method
    '''P-1''' is a [[:Category:Factorization|factorization method]] invented by John P ...a prime which does not divide the integer ''a'', then <math>a^{p-1}\equiv 1 \mbox{( mod } p)</math>.
    5 KB (814 words) - 01:35, 12 March 2019
  • Brent-Suyama extension
    ...ossibility of finding factors outside the normal range expected from the P-1 bounds. ...than B1. Then by [[Fermat's Little Theorem]], a prime number p | S-1 if p-1 | E.
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  • Self-initializing quadratic sieve
    :<math>45^2\,\equiv \,2^4*7^0*13^1</math> :<math>123^2\,\equiv \,2^{10}*7^0*13^1</math>
    10 KB (1,763 words) - 02:56, 12 March 2019
  • Special number field sieve
    :<math>\Theta\left(\exp\left( \left(\frac{32}{9}n\right)^{\frac{1}{3}} (\log n)^{\frac{2}{3}} \right)\right).</math>
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  • Fermat pseudoprimality test
    :<math>a^{p-1}\equiv 1\,\pmod{p}</math> ...', we perform <math>a^{N-1}\equiv 1\,\pmod{N}</math>. If the result is not 1, the number must be composite. Otherwise the number is either a prime or a
    1 KB (164 words) - 10:56, 6 February 2019
  • Miller-Rabin pseudoprimality test
    *The Fermat Little Theorem that states: <math>a^{p-1}\equiv 1\,\pmod p</math>. ...v 1\,\pmod p</math> then <math>m\equiv 1\,\pmod p</math> or <math>m\equiv -1\,\pmod p</math>.
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  • Andrew Thall
    ...[Lucas-Lehmer test]] for a [https://en.wikipedia.org/wiki/MasPar Maspar MP-1] (a [[Single instruction, multiple data|SIMD]] supercomputer released in 19
    2 KB (225 words) - 00:22, 15 January 2024
  • Generalized Fermat number
    ...+1}}-1)/(2^{p^n}-1)</math> where p is the prime of apparition rank r (r(2)=1, r(3)=2, r(5)=3, ...) and n is greater or equal to 0. :<math>F_{n,1}</math> generates the [[Fermat number]]s.
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  • Patrick Laroche
    ...very of the [[M51|51th known Mersenne prime]] 2<sup>{{Num|82589933}}</sup>-1.
    987 bytes (147 words) - 01:27, 15 January 2024
  • PrimeGrid
    ...ipants (on about 16,000 host computers) from 89 countries, reporting about 1,860 [[Computing power#FLOPS|teraflops]].<ref>[https://www.boincstats.com/st ...Riesel Problem|The Riesel Problem]]: helping to solve the [[Riesel problem 1|Riesel problem]].
    3 KB (458 words) - 10:28, 26 March 2024
  • Worktype
    *[[P-1_factorization_method|P-1]]: A second method that finds factors. *P-1 factoring: See P-1 above.
    4 KB (603 words) - 02:31, 18 August 2019
  • 321 Search
    ...d computing project]] to search for [[prime]]s of the form 3*2<sup>n</sup>-1.
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  • Sieve of Eratosthenes
    *First, list out all the integers <math>1 \leq k \leq N</math>. *<math>1</math> is not considered prime; it is ignored.
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  • Top GIMPS teams
    | 1 || [[PrimeNet]] || 32679708 *[https://www.mersenne.org/report_top_500_custom/?team_flag=1&type=0&rank_lo=1&rank_hi=50&start_date=1994&end_date= Lifetime team stats] at [[PrimeNet]]
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  • Aliquot sequence
    :<math>s_n = \sigma(s_{n-1}) - s_{n-1}</math> :<math>s_0\ =\ 10,\ \sigma(10)=1\ +\ 2\ +\ 5\ +\ 10</math>
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  • Abundant number
    The divisors of 12 are <math>(1, 2, 3, 4, 6, 12)</math>, so :<math>\sigma(12)\ =\ 1+2+3+4+6+12\ =\ 28</math>
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  • Prime
    ...''') is an [[integer]] greater than 1 that is only divisible by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. ...P divides evenly into Q, because each division would leave a remainder of 1. If Q is not prime, it must be evenly divisible by some prime larger than P
    2 KB (447 words) - 00:22, 10 July 2023
  • Twin prime
    ...of the reciprocals of all twin primes converges to Brun's constant (about 1.902160583104). | align="right" | 10<sup>1</sup> || align="right" | 2
    2 KB (255 words) - 06:08, 21 February 2023
  • Riesel Prime Search
    :The project is searching for [[Riesel prime]]s {{Kbn|k|n}}, {{Vk}} > 1. ...ght {{Vk}}-values that produce a very small number of primes (opposite to (1) above)
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  • Nomenclature and notation
    ...math> (the exponent) must also be prime. Thus, the notation of <math>2^{p}-1</math> is generally used when discussing the search for a [[Mersenne prime] ...mber that is itself prime '''and''' can be written in the form <math>2^{x}-1</math>. These are what [[GIMPS]] is searching for.
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  • Computing power
    ...calculating effort one 90 MHz Pentium computer produces over the course of 1 calendar year (365 days). Because of the early adoption of this unit by [[G ...ay. One P90 year equals 5.075 GHz-days. 1 TFLOPS equals 500 GHz-days, thus 1 P90 year is about 10 GFLOPS.
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  • Math rendering
    ...pm i-1} C</code> || <math> A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C</math> | <code>\overbrace{ 1+2+\cdots+100 }^{5050}</code> || <math>\overbrace{ 1+2+\cdots+100 }^{5050}</math>
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  • Seventeen or Bust/Primes
    | 4847 || {{PP|75994|4847 &times; 2<sup>{{Num|3321063}}</sup>+1}} || 999744 || 2005-10-15 || Richard Hassler | 5359 || {{PP|67719|5359 &times; 2<sup>{{Num|5054502}}</sup>+1}} || 1521561 || 2003-12-06 || Randy Sundquist
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  • Seventeen or Bust/FAQ
    *[[Seventeen or Bust/Factoring|Factoring]]: How can I P-1 factor for SoB? It's too complicated!
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  • Seventeen or Bust/Factoring
    ...actoring for Seventeen or Bust is made by the [[P-1 factorization method|P-1 factoring]] method only. The programs commonly used to P-1 factor these numbers are [[Prime95]] and [http://linux.redbird.com/~alien88
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  • Why participate in GIMPS?
    ...are known) are all closely related to the primes of the form 2<sup>p</sup>-1, for some prime ''p'' (now called [[Mersenne prime|Mersennes]]). So the que ...w. Mersennes have one of the simplest possible forms for primes, <math>2^p-1</math>. The proof of their primality has an elegant simplicity (to a mathem
    7 KB (1,252 words) - 09:47, 7 March 2019
  • Sieving
    ...etween 2 and the square root of the maximum member of the sequence, set to 1 those elements of the array that correspond to the multiples of the current ...composites. But this can be fixed easily by do not setting that element to 1 if the member of the sequence is equal to the prime used in the loop.
    3 KB (521 words) - 11:14, 6 February 2019
  • Sieving program
    ...ing sieving of generalized Cullen/Woodall numbers n &times; b<sup>n</sup>+-1) http://sites.google.com/site/geoffreywalterreynolds/programs/gcwsieve ...ieving for factors of numbers of the form K &times; 2<sup>n</sup> + 1 or - 1. Independent of K's, but good for many N's too) and [[TPSieve]] (similar to
    2 KB (220 words) - 11:42, 7 March 2019
  • NewPGen
    *k&times;b<sup>n</sup>+1 *k&times;b<sup>n</sup>-1
    3 KB (529 words) - 09:32, 7 March 2019
  • Sophie Germain prime
    .... For example, 23 is a Sophie Germain prime because it is a prime and 2*23+1 = 47, also prime. | 1 || 2 || -
    1 KB (171 words) - 04:26, 3 November 2020
  • M13
    '''M13''' is the 13th known [[Mersenne prime]] 2<sup>521</sup>-1 found on 1952-01-30 by [[Raphael M. Robinson]].
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  • M14
    '''M14''' is the 14th known [[Mersenne prime]] 2<sup>607</sup>-1 found on 1952-01-30 by [[Raphael M. Robinson]].
    420 bytes (48 words) - 14:49, 19 September 2021
  • M15
    '''M15''' is the 15th known [[Mersenne prime]] 2<sup>{{Num|1279}}</sup>-1 found on 1952-06-25 by [[Raphael M. Robinson]].
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  • M16
    '''M16''' is the 16th known [[Mersenne prime]] 2<sup>{{Num|2203}}</sup>-1 found on 1952-10-07 by [[Raphael M. Robinson]].
    431 bytes (49 words) - 14:49, 19 September 2021
  • M17
    '''M17''' is the 17th known [[Mersenne prime]] 2<sup>{{Num|2281}}</sup>-1 found on 1952-11-09 by [[Raphael M. Robinson]].
    431 bytes (49 words) - 14:49, 19 September 2021
  • M18
    '''M18''' is the 18th known [[Mersenne prime]] 2<sup>{{Num|3217}}</sup>-1 found on 1957-09-08 by [[Hans Riesel]].
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  • M11
    '''M11''' is the 11th known [[Mersenne prime]] 2<sup>107</sup>-1</math> found in 1914 by [[Ralph Ernest Powers]].
    412 bytes (47 words) - 14:23, 17 February 2019
  • M10
    '''M10''' is the 10th known [[Mersenne prime]] <math>2^{89}-1</math> found in 1911 by [[Ralph Ernest Powers]].
    403 bytes (44 words) - 13:54, 17 February 2019
  • Proth's theorem
    Let <math>p = k*2^n+1</math> and <math>k < 2^n</math>; then <math>p</math> is prime if there is a :<math>a^{(p-1)/2} \equiv -1\pmod{p}</math>.
    549 bytes (88 words) - 18:15, 28 September 2023
  • Proth prime
    Different from this definition all values ''n'' ≥ 1 are listed in {{SITENAME}}.
    656 bytes (91 words) - 07:02, 31 August 2020
  • Riesel prime
    Different from this definition all values {{Vn}} ≥ 1 are listed in {{SITENAME}}. ...servations/status"]: [https://www.mersenneforum.org/showpost.php?p=109831 #1 (2007-07-08)] - [https://www.mersenneforum.org/showpost.php?p=655641 #1081
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  • Riesel Sieve
    Riesel Sieve is a distributed effort to prove the [[Riesel problem 1|Riesel problem]] (which states that {{Vk}}=509203 is the smallest possible
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  • M26
    '''M26''' is the 26th known [[Mersenne prime]] 2<sup>{{Num|23209}}</sup>-1 found on 1979-02-09 by [[Landon Curt Noll]].
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  • Power of two
    ...1 kibibyte. 1024<sup>2</sup> bytes = 1 mebibyte, 1024<sup>3</sup> bytes = 1 gibibyte, and so on.
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  • Introductory stuff
    :Test=13974239,65,1 ...= 2<sup>2</sup> - 1 is a Mersenne prime; so is 7 = 8 - 1 = 2<sup>3</sup> - 1. See [[Mersenne prime]].
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  • What use are prime numbers anyway?
    ...orem of Arithmetic", states that every positive integer (except the number 1) can be expressed in exactly one way as the product of one or more primes.
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  • Glossary
    *'''[[Fermat number]]''' - Numbers of the form <math>2^{2^n} + 1</math>. *'''[[Mersenne prime]]''' - Primes of the type <math>2^n-1</math> (implying n is also prime).
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  • Overclocking and Prime95
    ...If you proved that the maximum accumulated error cannot be more than +/-0.1 on a correct calculation then a processing error must have occurred. ...ng around the TLC site. Get things working stable and then drop the FSB by 1 MHz to give you a better safety factor.
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  • Overclocking
    ...de Bus' (hereafter referred to as the FSB) speed. Lets say you have a nice 1.6A P4. (Historically this was one of the best CPUs for overclocking). That ...omething called a multiplier. It just so happens that the multiplier for a 1.6GHz P4 is 16. The reason behind this is quite simple...
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  • Triangular number
    :4△ = 4 + 3 + 2 + 1 = 10 (10 pin bowling uses a triangular arrangement.) :5△ = 5 + 4 + 3 + 2 + 1 = 15 (a common billiards arrangement is 15 balls in a triangle.)
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  • Pocklington's theorem
    ...= 1</math>, then each prime factor ''q'' of ''N'' has the form <math>q^kr+1</math>. ...it can be deduced a primality test when only a partial factorization of N-1 is known:
    2 KB (346 words) - 19:51, 30 August 2019
  • Multiple polynomial quadratic sieve
    :<math>45^2\,\equiv \,2^4*7^0*13^1</math> :<math>123^2\,\equiv \,2^{10}*7^0*13^1</math>
    6 KB (1,068 words) - 14:33, 13 February 2019
  • Proth number
    ...condition that 2<sup>{{Vn}}</sup> > {{Vk}}, all odd integers greater than 1 would be Proth numbers, but most pages lists them, too.
    670 bytes (104 words) - 10:59, 9 July 2021
  • 27121 Search
    ...f the form: 27 &times; 2<sup>n</sup> ± 1 and 121 &times; 2<sup>n</sup> ± 1. ...es are 27 &times; 2<sup>1902689</sup>-1 and 27 &times; 2<sup>2218064</sup>+1, which weighs in at 572768 digits and 667706 digits, found on 2009-12-10 a
    983 bytes (138 words) - 13:25, 8 February 2019
  • 12121 Search
    ...2<sup>n</sup>-1, it also searches for primes of 121 &times; 2<sup>n</sup>+1 after being a partner of [[PrimeGrid]]. ...The Prime Pages]], their largest number is 121 &times; 2<sup>2033941</sup>-1 which weighs in at 612280 digits. This prime was found on 2006-01-28.
    984 bytes (143 words) - 13:30, 8 February 2019
  • Assignments
    ==How is CPU years calculated for P-1, and LL test?== *'''P-1 credit'''
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  • GIMPS type of work
    ==P-1 factoring== ...d. The more memory available the greater the chance of finding a factor. P-1 factoring takes longer per single unit than TF, but the chance of finding a
    4 KB (757 words) - 15:17, 25 July 2020
  • How to get started with GIMPS
    ==Step 1== ...ticly linked ''sprimexxx.tar.gz'' for systems that do not have the glibc 2.1 runtime libraries.
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  • GIMPS networking, PrimeNet and communication
    ...58 days and set the network retry to 300 minutes. Set the cache option to 1 day to prevent the client from attempting to get more work. ...cure proxy, the proxy password is encoded and a new parameter '''ProxyMask=1''' set. To change the password, simply change the ProxyPass= value, and eit
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  • GIMPS chances of finding a prime number
    ...ast fifty throws, the chances of you throwing a six on your next throw are 1:6. What has happened in the past does not affect the number of faces on the ...; there are only two possible outcomes, so the chance of it being heads is 1:2. From the fact that a number either is prime or it is not one might suppo
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  • Worktodo.txt
    ...t ID]],[[exponent]],how far factored,has been [[P-1 factorization method|P-1]]'ed ...ble check|DoubleCheck]]=assignment ID,exponent,how far factored,has been P-1'ed
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  • Bit level
    ...e 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 for factors from one bit level to | align="right" | 1 0000 1001 0011 0010
    1 KB (193 words) - 18:47, 14 December 2023
  • P+1 factorization method
    ...Hugh Williams in 1982 and it is based in the [[p-1 factorization method|p-1]] method. <math>\large U_0 = 0\,,\, U_1 = 1\,,\, V_0 = 2\,,\, V_1 = u </math>
    8 KB (1,536 words) - 11:35, 12 February 2019
  • Woodall number
    :W<sub>n</sub> = n &times; 2<sup>n</sup>-1
    374 bytes (59 words) - 16:41, 31 August 2021
  • SNFS polynomial selection
    ...a^{59}-1</math> number currently not completely factored is <math>208^{59}-1</math>, the polynomials would be <math>x^5-208</math> on the algrebraic sid ...r polynomials are, for example for <math>N=208^{61}-1</math>, <math>208x^5-1</math> and <math>x-6557827967253220516257857536</math>.
    7 KB (1,238 words) - 16:14, 12 February 2019
  • NFSNET
    |<math>5^{311}+1</math> || 13132762900451821968706840158108829466847315743095478589617724372 |<math>5^{313}-1</math> || 21428622089774767159447145142284385968882142917892658511907216761
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  • Mersenneplustwo factorizations
    ...p>-1 being prime). All such numbers are divisible by 3 since 2<sup>p</sup>-1 is not divisible by 3 (it's assumed to be prime) and 2<sup>p</sup> is not d
    2 KB (215 words) - 13:33, 5 March 2019
  • ECMclient
    1. Get it [http://home.earthlink.net/~elevensmooth/ElevenSmooth.zip here] and
    2 KB (383 words) - 11:16, 26 February 2019
  • GMP-ECM
    ...d|ECM]], [[P-1 factorization method|p-1]] and [[P+1 factorization method|p+1]] algorithms. The current version is 7.0.5-dev (svn 3038). 1. First you'll need to get the right environment and stuff to work with:
    4 KB (567 words) - 10:54, 6 December 2019
  • Gerbicz error checking
    # <math>d(t+1) \equiv d(t)*u((t+1)L) \pmod{p}</math> # <math>d(t+1) \equiv u(0)*d(t)^{2^L} \pmod{p}</math>
    3 KB (528 words) - 14:59, 3 October 2023
  • Jacobi error checking
    ...the residue plus 2 <math>\left(\frac{Res+2}{M_p}\right)</math> has to be +1 ...the residue minus 2 <math>\left(\frac{Res-2}{M_p}\right)</math> has to be -1
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  • ECPP-DJ
    |latest=1.04<br><small>2014-08-16</small> :BLS75 proof using N-1
    2 KB (237 words) - 18:04, 2 December 2019
  • BLS75
    ...5-0384673-1.pdf "New Primality Criteria and Factorizations of 2^{{V|m}} ± 1"]. ''Mathematics of Computation.'' Volume 29, Number 130: 620-647. ...-1 factorization method|{{V|p}}-1]] and [[P+1 factorization method|{{V|p}}+1]].
    584 bytes (85 words) - 20:12, 26 October 2020
  • GIMPS PrimeNet reports
    ...Lehmer test|LL Tests]], LL [[double check]]s, [[P-1 Factorization methid|P-1]] factoring, and [[Elliptic curve method|ECM Factoring]]. These are also up Lists the current [[bit level]] and [[P-1 factorization method|P-1]] [[bounds]] for a given list of exponents which have no known factors and
    7 KB (1,137 words) - 15:30, 13 August 2020
  • Sierpiński-Riesel Base 5
    ...base 5]]: There are <b>{{#expr:{{PAGESINCATEGORY:PrimeGrid Riesel base 5}}-1}}</b> values to prove. ...5]]: There are <b>{{#expr:{{PAGESINCATEGORY:PrimeGrid Sierpiński base 5}}-1}}</b> values to prove.
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  • M27
    | foundwith=[[Lucas-Lehmer test]] / [[Cray 1]]
    302 bytes (26 words) - 08:16, 18 February 2019
  • M28
    | foundwith=[[Lucas-Lehmer test]] / [[Cray 1]]
    278 bytes (23 words) - 08:22, 18 February 2019
  • History
    :'''2018-12-07''' : '''[[M51]]''' = 2<sup>{{Num|82589933}}</sup>-1 is reported prime. :'''2017-12-26''' : '''[[M50]]''' = 2<sup>{{Num|77232917}}</sup>-1 is reported prime.
    3 KB (479 words) - 10:55, 7 March 2019
  • System requirements and compatible hardware
    ...4-bit SP 1 / 64-bit SP 2 / AMD 64-bit / Itanium 64-bit / Itanium 64-bit SP 1 / Itanium 64-bit SP 2) :Windows NT (3 / 4 / 4 SP 1 / 4 SP 2 / 4 SP 3 / 4 SP 4 / 4 SP 5 / 4 SP 6)
    766 bytes (95 words) - 12:25, 19 February 2019
  • Mersenne composite
    A '''Mersenne composite''' is any number of the form 2<sup>n</sup>-1 which is a [[composite number]]. ...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
    *'''P-1''': [[P-1 factorization method]] assignments. These exponents have had a reasonable a ...lows TF. (Currently the sequence is TF to final [[bit level]] minus one, P-1, then final bit level TF).
    7 KB (1,072 words) - 13:10, 19 February 2019
  • Square number
    :1<sup>2</sup> = 1 ...>2</sup> equals to the sum of the first ''n'' odd numbers (<math>n^2 = 2(n-1)^2-(n-2)^2+2</math>). A square number is also the sum of two consecutive [[
    3 KB (408 words) - 13:56, 19 February 2019
  • Editing
    ...wiki>"The [[Sierpiński problem]] is about numbers of the form <math>k*2^n+1</math>"</nowiki> : "The [[Sierpiński problem]] is about numbers of the form <math>k*2^n+1</math>"
    5 KB (805 words) - 06:50, 1 May 2019
  • Riesel number base 5
    Using the same method presented in the [[Riesel problem 1|Riesel problem]] article, it was found that {{Kbn|-|346802|5|n}} is multipl
    589 bytes (90 words) - 10:30, 26 March 2024
  • Riesel problem 1
    Currently, there are '''{{#expr:{{PAGESINCATEGORY:Riesel problem 1|pages|R}}-2}}''' {{Vk}}-values smaller than {{Num|509203}} that have no kno ...with {{Vn}} in the interval 2<sup>{{V|m}}</sup> ≤ {{Vn}} < 2<sup>{{V|m}}+1</sup>. <ref>[http://www.prothsearch.com/rieselprob.html Riesel problem] by
    6 KB (689 words) - 18:14, 4 April 2024
  • Tutorial LLR
    ...umbers are of the same form (normally k*2<sup>n</sup>+1 or k*2<sup>n</sup>-1).
    2 KB (337 words) - 13:24, 20 February 2019
  • Peano postulates
    ...n 1889; it characterises the set (class, condition) of [[natural number]]s 1, 2, 3, etc., and consists of the following '''Peano postulates''' (also cal #''1'' is a natural number
    2 KB (269 words) - 17:04, 24 October 2020
  • LL test distribution
    ...senne number successfully double checked was 2<sup>{{Num|666666667}}</sup>-1, with {{Num|200686664}} digits. It was tested and double checked by LaurV.
    2 KB (415 words) - 21:32, 12 February 2020
  • ElevenSmooth
    ...actors of the [[Mersenne number]] M(3326400) = 2<sup>{{Num|3326400}}</sup>-1.
    366 bytes (42 words) - 12:49, 21 February 2019
  • Factoring report
    ...ports, it lists the current [[bit level]] and [[P-1 factorization method|P-1]] [[bounds]] for a given list of exponents, that have no known factors.
    524 bytes (78 words) - 13:12, 21 February 2019
  • Mersenne.ca
    ...of finding a [[Mersenne prime]], many GIMPS users purposely skipped the P-1 stage to spend more time doing L-L first-time tests. ...sts are graphed, with known factors overlaid to show how close any known P-1 test(s) were to finding the factor(s). A similar graph shows known factors
    9 KB (1,396 words) - 15:42, 25 February 2019
  • Value k
    :<math>2kp+1</math> where <math>p</math> is the [[exponent]] in <math>2^p-1</math>. <math>\begin{align}2^{23}-1 &= 8388607\\&= 47 * 178481\\
    702 bytes (99 words) - 10:43, 25 October 2020
  • Aurifeuillian factor
    *Numbers of the form <math>2^{4k+2}+1</math> have the following '''Aurifeuillian factorization''': [http://mathwo ::<math>2^{4k+2}+1 = (2^{2k+1}-2^{k+1}+1)\cdot (2^{2k+1}+2^{k+1}+1)</math>
    10 KB (1,257 words) - 08:04, 24 June 2019
  • Cullen number
    {{V|C<sub>n</sub>}} is [[prime]] for {{Vn}} = 1, 141, {{PP|40087|4713}}, {{PP|38112|5795}}, {{PP|37374|6611}}, {{PP|23436|1
    2 KB (252 words) - 17:39, 31 August 2021
  • Factoring program
    *[[GMP-ECM]] (performing ECM, P-1, and P+1) http://gforge.inria.fr/projects/ecm/ ...ime|MPrime]] (for Linux , FreeBSD, and MacOSX) (performing ECM, P-1, and P+1 to search for factors of a &times; b<sup>n</sup> ± c) ftp://mersenne.org/g
    2 KB (273 words) - 21:53, 8 July 2023
  • Mtsieve
    *dmdsieve: search for factors of numbers of the form 2*k*(2<sup>p</sup>-1)+1 (potential divisors of [[Double Mersenne number]]s)
    2 KB (338 words) - 06:58, 28 March 2023
  • Carol-Kynea prime
    ...numbers''' are numbers of the form <math>(b^n-1)^2-2</math> and <math>(b^n+1)^2-2</math>, respectively, while '''Carol primes''' and '''Kynea primes''' ...h>(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]] which has one of the above
    8 KB (1,172 words) - 00:38, 6 July 2023
  • Multifactorial prime
    ...r]] which is prime and of the form <math>\ n!_2{±}1,\ n!_3{±}1,\ n!_4{±}1</math>, and so on.
    429 bytes (50 words) - 14:29, 25 March 2019
  • Multifactorial number
    :<math>n! = 1 \cdot 2 \cdot 3 \cdots (n{-}2) \cdot (n{-}1) \cdot n</math> for <math>n \geq 1</math>.
    560 bytes (81 words) - 14:36, 20 July 2021
  • Riesel prime 2 1
    |Rk=1 2;T:ST;C:'''[[M1]]''', {{NWo|+|1}}, {{NWo|-|2}}, {{NWo|4|1}}
    2 KB (288 words) - 11:41, 3 April 2023
  • Riesel prime 2 3
    1;T:ST 2;T:ST;C:{{NWo|+|2}}, {{NWi|MM|4|1}}
    5 KB (523 words) - 09:47, 5 October 2023
  • Riesel prime 2 5
    {{HistC|2021-03-21|1-1000000|Gary Barnes|574248}}, double check, results included
    2 KB (196 words) - 18:36, 9 October 2021
  • Riesel prime 2 7
    1 {{HistC|2021-03-21|1-1000000|Gary Barnes|574248}}, double check, results included
    3 KB (264 words) - 22:08, 5 July 2023
  • Riesel prime 2 15
    1;T:ST {{HistC|2021-03-21|1-1000000|Gary Barnes|574248}}, double check, results included
    2 KB (252 words) - 13:29, 5 May 2024
  • Riesel prime 2 2145
    1
    3 KB (329 words) - 07:59, 17 August 2021
  • Factorial prime
    A '''Factorial prime''' is a [[prime]] of the form '''[[Factorial number]] ± 1'''. *Factorial primes of the form ''n!-1'' are of the {{OEIS|l|A002982}}.
    488 bytes (70 words) - 07:11, 1 April 2019
  • Carol-Kynea prime 50
    1
    490 bytes (45 words) - 12:30, 3 July 2019
  • Williams prime MM 10
    1
    1 KB (143 words) - 02:41, 5 November 2020
  • Williams prime PM 2
    |WiNlist={{Reuse Primelist|Riesel prime 2 3|RNlist|1}}
    246 bytes (35 words) - 14:04, 1 August 2021
  • Williams prime MP 2
    |WiMaxn={{GP|Proth prime 2 1|PMaxn}} |WiDate={{GP|Proth prime 2 1|PDate}}
    243 bytes (35 words) - 08:06, 1 August 2021
  • Williams prime PP 2
    |WiNlist={{Reuse Primelist|Proth prime 2 3|PNlist|1}}
    259 bytes (36 words) - 10:52, 13 July 2021
  • Williams prime MM 3
    |WiNlist={{Reuse Primelist|Riesel prime 3 2|RNlist|1}}
    248 bytes (35 words) - 09:56, 16 March 2023
  • Williams prime PM 3
    |WiNlist={{Reuse Primelist|Riesel prime 3 4|RNlist|1}}
    248 bytes (35 words) - 13:40, 16 March 2023
  • Williams prime MM 2
    |WiMaxn={{GP|Riesel prime 2 1|RMaxn}} |WiDate={{GP|Riesel prime 2 1|RDate}}
    244 bytes (35 words) - 20:47, 31 July 2021
  • Williams prime MM 6
    1
    642 bytes (56 words) - 09:01, 22 November 2023
  • Williams prime MM 7
    1
    430 bytes (42 words) - 11:00, 21 May 2019
  • Williams prime MM 9
    1
    412 bytes (45 words) - 20:59, 31 July 2021
  • Williams prime MM 11
    1
    366 bytes (36 words) - 20:54, 31 July 2021
  • Williams prime MM 12
    1
    318 bytes (29 words) - 11:41, 21 May 2019
  • Williams prime MM 13
    {{HistC|2019-04-17|100000|Dylan Delgado}}, double checked {{Vn}} = 1-5000
    394 bytes (38 words) - 20:54, 31 July 2021
  • Williams prime MM 14
    1 ...://ostracodfiles.com/primes14/primes.php "PRIME NUMBERS OF THE FORM A*14^B-1"]
    402 bytes (44 words) - 11:53, 21 May 2019
  • Williams prime MM 17
    1
    214 bytes (19 words) - 12:20, 21 May 2019
  • Williams prime MM 20
    1
    199 bytes (19 words) - 12:28, 21 May 2019
  • Williams prime MM 21
    1
    231 bytes (19 words) - 12:53, 21 May 2019
  • Williams prime MM 22
    1
    221 bytes (19 words) - 13:00, 21 May 2019
  • Williams prime MM 25
    1
    231 bytes (19 words) - 13:05, 21 May 2019
  • Riesel prime 2 31
    1 {{HistC|2021-03-21|1-1000000|Gary Barnes|574248}}, double check, results included
    2 KB (228 words) - 19:30, 19 July 2023
  • Williams prime MM 27
    1
    282 bytes (26 words) - 20:58, 31 July 2021
  • Williams prime MM 29
    1
    207 bytes (19 words) - 13:21, 21 May 2019
  • Williams prime MM 31
    1
    235 bytes (19 words) - 13:26, 21 May 2019
  • Williams prime MM least
    {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b-1)|b|n}}, least {{Vn}}-values}} ...{{Vb}} ≤ 2049 which generates a [[Williams prime]] of the form {{Kbn|(b-1)|b|n}}.
    2 KB (211 words) - 07:40, 1 August 2021
  • Proth prime 2 1
    |Pk=1 1
    212 bytes (30 words) - 15:35, 2 October 2022
  • Proth prime 2 3
    1;T:GT
    3 KB (336 words) - 16:58, 15 April 2024

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