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  • Williams prime
    ...a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1. ...Smallest <ref>The list of smallest primes of any base is an ASCII file for 2 ≤ ''b'' ≤ 1024. For unknown values only the base is given.</ref> !! Rem
    5 KB (744 words) - 07:30, 5 August 2019
  • Carol-Kynea prime 2
    |CKBase=2 2
    5 KB (537 words) - 08:17, 9 October 2020
  • Carol-Kynea prime 6
    2 2
    1 KB (85 words) - 10:45, 16 April 2023
  • Srsieve
    '''srsieve''' (and '''sr1/2/5sieve''') is used to create sieve files for one or more sequences. Those s
    2 KB (265 words) - 07:36, 28 May 2021
  • Examples
    <pre><math>\sideset{_1^2}{_3^4}\prod_a^b</math></pre> :<math>\sideset{_1^2}{_3^4}\prod_a^b</math>
    11 KB (1,236 words) - 14:41, 3 September 2020
  • Great Internet Mersenne Prime Search
    ...as born in 1588. 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 ...n P<sub>n</sub>''': 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
    ...is one less than a [[power of two]]. For example, 3 = 4 &minus; 1 = {{Kbn|2}} is a Mersenne prime; so is 7 = 8 &minus; 1 = {{Kbn|3}}. On the other hand :<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]]. ...senne number <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
    ...for finite differences <math>\large\delta y</math> and <math>\large\delta^2 y</math> and many more.
    16 KB (2,614 words) - 11:48, 14 January 2024
  • Marin Mersenne
    : <math>\sqrt[4]{\frac{2}{3-\sqrt{2}}}</math> ...ined if the ratio of the air oscillation frequencies is also 1&nbsp;:&nbsp;2, which in turn is consistent with the source-air-motion-frequency-equivalen
    11 KB (1,582 words) - 01:17, 15 January 2024
  • Euclid
    ...c and fundamental parts of geometry; 1. A point is that which has no part. 2. A line is breadthless length. From these definitions of the point and the
    2 KB (341 words) - 11:43, 14 January 2024
  • Landon Curt Noll
    ...l candidates with prime [[exponent]]s up to and including M26. During that 2 hour period, it duplicated what had previously occurred over a 520 year per
    2 KB (333 words) - 12:40, 9 February 2022
  • Laura A. Nickel
    ...aura 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 internatio
    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>1</sub> = {{Kbn|+|2}} = 5
    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/pr ...glund also confirmed using Mlucas running on an [[Amazon EC2]] instance in 2.7 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.o ...ill, who each ran the [[CUDALucas]] software on NVidia Titan Black GPUs in 2.3 days
    2 KB (283 words) - 11:50, 18 February 2019
  • Curtis Cooper
    ...-09-04 when the pair discovered the [[M44|44th]] known [[Mersenne prime]], 2<sup>32 582 657</sup>-1. As of 2008-09-15 his account is ranked #1 on [[Prim On 2013-01-25 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
    ...k's father enabled them to find factors for all integers up to ''48911<sup>2</sup>'', and these factorisations were published in a book in 1929. That su ...improvement to the [[Lucas primality test]] for [[Mersenne prime]]s <math>2^p{-}1</math>, extending its application to all odd prime exponents ''p'', a
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Home prime
    ==Base 2== HP<sub>2</sub>(10):
    980 bytes (143 words) - 13:22, 6 March 2019
  • Base 2 Home Prime Results
    ==Search for Home Primes base 2== {{HP|2|2295|281|189|2013-07-23|}}
    2 KB (175 words) - 18:45, 14 December 2023
  • Cunningham tables
    *[[2 Minus Tables]] *[[2 Plus Tables]]
    614 bytes (69 words) - 12:08, 13 February 2019
  • Cunningham project
    ...zation]] 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 exponent <math>n</math> are sele | 2 || 3 || 5 || 6 || 7 || 10 || 11 || 12
    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. .... They were testing this theory 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
    ...S<sub>0</sub>=4 and S<sub>n</sub> = (S<sub>n-1</sub>)<sup>2</sup> &minus; 2. In 1930, the American mathematician [[Derrick Henry Lehmer]] (1905 - 1991) ...hat 2<sup>5</sup> - 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
    ...te the [[Lucas-Lehmer test]]. In 1876, Lucas proved the primality of <math>2^{127}{-}1</math> ([[M12]]) and this remained the highest [[Mersenne prime]]
    2 KB (296 words) - 01:09, 15 January 2024
  • Decimal
    ...at people use in most of the world. [[Computer]]s use '''binary''' or base 2. The length of a number (how many [[digit]]s it takes to write the number) ...us 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. The
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    Different bases are often used in [[computer|computers]]. Binary (base 2) is used because at the most simple level, computers can only deal with 0s ! Decimal<br>base=10 !! Binary<br>base=2 !! Hexadecimal<br>base=16
    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
    ...[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 usually den ...f [[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
  • Mathematics
    ...s <math>a</math> and <math>b</math>, <math>(a + b) * (a - b)\,=\,a^2\,-\,b^2</math>.
    1 KB (186 words) - 17:00, 5 February 2019
  • Addition
    In [[mathematics]]: to sum 2 numbers. It is normally symbolized by the plus sign '+'. :2 + 2 = 4
    333 bytes (43 words) - 16:55, 29 August 2022
  • Multiplication
    ...largest available register is ''n'' bits wide the factors can only be ''n/2'' bits wide each.
    2 KB (271 words) - 17:00, 29 August 2022
  • Factorial number
    :<math>n! = 1 \cdot 2 \cdot 3 \cdots (n{-}2) \cdot (n{-}1) \cdot n</math> :5! = 5 * 4 * 3 * 2 * 1 = 120
    729 bytes (93 words) - 13:40, 5 November 2023
  • Division
    :<math>\frac 63 = 2</math> :<math>2 \times 3 = 6</math>.
    2 KB (368 words) - 16:58, 29 August 2022
  • Remainder
    ...ft over after dividing something. Dividing 13 into groups of 5 would yield 2 groups, with a remainder of 3.
    245 bytes (34 words) - 14:07, 18 January 2019
  • 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
    ...composite numbers have small factors (half of the numbers are multiples of 2, a third are multiples of 3 and so on) it pays to run factorization methods
    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.o *[https://www.youtube.com/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
    ...found. [[M45]], [[M46]], and M47 were discovered in the order of M47, M45 (2 weeks later), then M46 (8 months later). On 2018-04-08 all tests below 2<sup>{{Num|43112609}}</sup>-1 were verified by [[GIMPS]], officially making
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    ...fafaf; background-color: #f9f9f9; border-collapse: collapse;" cellpadding="2" !Comparison of CPU core power!!Frequency!!Cores!!colspan="2" | [[Fast Fourier transform|FFT]]!![[Trial factoring]]!!TDP
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Even number
    ...n the physcial world, an even number of objects can be placed into exactly 2 groups that have the identical number of objects. All numbers ending in 0, 2, 4, 6, or 8 are even.
    425 bytes (61 words) - 11:19, 7 March 2019
  • Odd number
    An '''odd number''' is any [[integer]] that is not divisible by 2. ...notation, the odd numbers end in 1, 3, 5, 7 or 9. All prime numbers except 2 are odd.
    316 bytes (42 words) - 11:21, 7 March 2019
  • M41
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2.4 GHz Pentium 4 [[Personal computer|PC]] '''M41''' is the short hand used to refer to the 41st [[Mersenne prime]] 2<sup>{{Num|24036583}}</sup>-1.
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2 GHz Pentium 4 [[Personal computer|PC]] '''M43''' is the short hand used to refer to the 43rd [[Mersenne prime]] 2<sup>{{Num|30402457}}</sup>-1.
    1 KB (191 words) - 11:31, 18 February 2019
  • M46
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 3 GHz Core 2 [[Personal computer|PC]] '''M46''' is the short hand used to refer to the 46th [[Mersenne prime]] 2<sup>{{Num|42643801}}</sup>-1.
    2 KB (248 words) - 11:45, 18 February 2019
  • Riesel prime 2 21143
    |Rb=2
    403 bytes (26 words) - 18:40, 2 January 2023
  • Riesel prime 2 102765
    |Rb=2
    935 bytes (70 words) - 18:56, 10 December 2022
  • Riesel prime 2 182627
    |Rb=2
    255 bytes (24 words) - 22:20, 10 December 2022
  • Proth prime 2 1089927
    |Pb=2
    432 bytes (32 words) - 13:38, 2 January 2023
  • Proth prime 2 12321
    |Pb=2
    498 bytes (31 words) - 13:34, 2 January 2023
  • Riesel prime 2 187605
    |Rb=2 2;T:S
    490 bytes (35 words) - 12:22, 11 December 2022
  • Riesel prime 2 132165
    |Rb=2 2
    557 bytes (25 words) - 12:35, 11 December 2022
  • Proth prime 2 1943873
    |Pb=2
    334 bytes (32 words) - 15:12, 27 January 2023
  • Program
    ...Programming, Volume 2, 3rd Edition, 1997, Addison-Wesley, ISBN 0-201-89684-2
    2 KB (263 words) - 11:53, 7 February 2019
  • M42
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2.4 GHz Pentium 4 [[Personal computer|PC]] '''M42''' refers to the 42nd [[Mersenne prime]] 2<sup>{{Num|25964951}}</sup>-1.
    934 bytes (118 words) - 11:26, 18 February 2019
  • M1
    | nvalue= 2
    193 bytes (19 words) - 13:43, 17 February 2019
  • Conjectures 'R Us
    ...t in proving the [[Liskovets-Gallot conjectures]] for the forms {{Kbn|±|k|2|n}} where {{Vn}} is always odd '''and''' where {{Vn}} is always even. ==Sub-project #2==
    3 KB (503 words) - 02:20, 1 May 2024
  • Fast Fourier transform
    :<math>\large f_j = \sum_{k=0}^{n-1} x_k e^{-{2\pi i \over n} jk } \qquad j = 0, ... ,n-1.</math> Evaluating these sums directly would take O(''n''<sup>2</sup>) arithmetical operations . An FFT is an algorithm to compute the same
    17 KB (2,684 words) - 18:50, 28 September 2023
  • GpuLucas
    ...rform nearly two times faster than [[CUDALucas]] due to using non-power-of-2 [[Fast Fourier transform|FFT]] lengths. [http://www.mersenneforum.org/showt
    2 KB (239 words) - 11:12, 13 February 2019
  • GpuOwL
    ...ehmer test|LL]], [[Probable prime|PRP]]|title=gpuOwL|release=2017|latest=7.2<br>2020-11-01}} ...nch] at GitHub (version 1 uses 4M FFT and is about 50% faster than version 2) [http://www.mersenneforum.org/showpost.php?p=479585&postcount=320]
    1 KB (216 words) - 05:22, 1 December 2020
  • M32
    | foundwith=[[Lucas-Lehmer test]] / Maple on Harwell Lab [[Cray-2]] :2<sup>756 839</sup>-1, a number {{Num|227832}} [[decimal]] [[digit]] long was
    2 KB (279 words) - 08:35, 18 February 2019
  • M33
    '''M33''' refers to 33rd [[Mersenne prime]] number 2<sup>{{Num|859433}}</sup>-1. ...percomputer]]. Computation of [[Lucas-Lehmer test]] for this number took 7.2 hours.
    814 bytes (97 words) - 08:38, 18 February 2019
  • M34
    ...size (smallest to largest) and in order of discovery. Specifically M34 is 2<sup>{{Num|1257787}}</sup>-1, which is a number {{Num|378632}} [[decimal]] [
    3 KB (513 words) - 08:42, 18 February 2019
  • No Prime Left Behind
    ...2008-01-10. The project searches for [[Riesel prime]]s of the form {{Kbn|k|2|n}} with odd {{Vk}} and 300 < {{Vk}} < 1001 and {{Vn}} > 260000 not reserve
    745 bytes (111 words) - 02:17, 1 May 2024
  • 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
  • Absolute value
    <math>|z| = \sqrt{x^2+y^2}</math>
    556 bytes (89 words) - 16:58, 29 August 2022
  • Rational number
    ...power of 2 multiplied by a perfect power of 5, i.e. it has the form <math>2^n \times 5^m</math>.
    3 KB (541 words) - 15:01, 26 March 2023
  • Irrational number
    ...at to be a definition. Some examples of irrational numbers are <math>\sqrt{2}</math> or <math>e</math>.
    763 bytes (124 words) - 15:14, 26 March 2023
  • Binary
    ...umbers using only two [[digit]]s (usually, 0 and 1). Thus it is a [[base]] 2 numbering system. Example: 10110011<sub>2</sub> = 179<sub>10</sub>
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...eing the number referred to as "unity") numbers. 111 is a repunit, in base 2 it is equal to 7 (base 10), in base 3 it is equal to 13 (base 10). A '''Generalized repunit''' for any base {{Vb}} &ge; 2 is defined as
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    ...hen a single [[processor]], multiple processors, or multiple cores perform 2 or more operations (similar or different) at once, that is '''parallel comp ! Step !! Input 1 !! Operation !! Input 2 !! Result !! 1440<br>x 365
    3 KB (416 words) - 06:47, 1 May 2019
  • Long multiplication
    ...0. Computers normally use a very similar 'shift and add' algorithm in base 2. [[Prime95]] does not use this form of multiplication for large numbers, us
    2 KB (165 words) - 17:01, 29 August 2022
  • Trial factoring
    ...to use as trial divisors. If P(i) is the i'th prime number so P(1) = 2, P(2) = 3, P(3) = 5, etc, then the last prime factor possibility for some number ...< \sqrt{N}</math>) there is no need to try 7 since 2*7 is excluded because 2 will have been tried, 3*7 is excluded because 3 will have been tried, and 5
    7 KB (1,221 words) - 13:20, 11 February 2019
  • M40
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2 GHz Dell Dimension ...hort hand used to refer to the 40th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|20996011}}</sup>-1. This number was discovered to be [[prime]] on
    1 KB (189 words) - 11:17, 18 February 2019
  • Michael Shafer
    '''Michael Shafer''' discovered 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
    ...nt. He is credited with discovery of the [[M41|41st known Mersenne prime]] 2<sup>{{Num|24036583}}</sup>-1.
    695 bytes (93 words) - 11:46, 14 January 2024
  • M39
    ...hort hand used to refer to the 39th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|13466917}}</sup>-1. This number was discovered to be [[prime]] on
    868 bytes (109 words) - 11:14, 18 February 2019
  • M44
    ...efer to the 44th [[Mersenne prime]]. Currently that designation belongs to 2<sup>{{Num|32582657}}</sup>-1.
    997 bytes (129 words) - 11:35, 18 February 2019
  • M45
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2.83 GHz Core 2 Duo [[Personal computer|PC]] '''M45''' normally refers to 2<sup>{{Num|37156667}}</sup>-1, the 45th [[Mersenne prime]] in order of size
    2 KB (251 words) - 11:40, 18 February 2019
  • Odd Magnar Strindmo
    ...way who discovered the [[M46|46th Mersenne prime]] (chronologically 47th), 2<sup>{{Num|42643801}}</sup>-1. Strindmo goes by the alias '''Stig M. Valstad Strindmo's 3 GHz Core 2 Duo PC first reported the prime to GIMPS on 2009-04-12. However, due to a s
    991 bytes (141 words) - 00:33, 15 January 2024
  • Richard Crandall
    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
    | top5000id=2 ...hort hand used to refer to the 38th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|6972593}}</sup>-1. This number was discovered to be [[prime]] on
    1 KB (165 words) - 11:10, 18 February 2019
  • Nayan Hajratwala
    ...ers employee from Michigan who discovered the [[M38|38th Mersenne prime]], 2<sup>{{Num|6972593}}</sup>-1.
    809 bytes (109 words) - 23:55, 14 January 2024
  • Coprime
    ...s 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
  • Modular arithmetic
    ...s 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>. ...</b>/n<b>Z</b> of numbers modulo n contains the numbers 0, 1, 2, 3, ..., n-2 and n-1. The following operations are defined:
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    ...math>ab\,\equiv \,c\,\pmod{m}</math>. We will also assume that <math>m\,<\,2^n</math>. :<math>a'=2^n\,a\,\bmod{m}</math>.
    4 KB (582 words) - 17:01, 29 August 2022
  • Elliptic curve method
    ...iplication|multiplying]] lots of different prime numbers together. So that 2 x 3 x 5 x 7 x 11 x 13 etc will be a highly composite number. But that is on ...,9</math> is a quadratic expression (because the highest power of ''x'' is 2).
    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.
    2 KB (224 words) - 11:00, 18 February 2019
  • University of California, Los Angeles
    ...ter]]. Robinson's Mersenne primes were the first to be found in 75 years (2 in the very first day of the run, no less). And he raised the number of dig ...ugust of 2008, a Dell Optplex 745 (running a Intel Core 2 Duo E6600 CPU at 2.4GHz) in the UCLA Math department computer lab, found [[M47|47th Mersenne p
    2 KB (347 words) - 14:54, 19 September 2021
  • Raphael M. Robinson
    ...e [[Mersenne number]]s were all composite except for 17 values of ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, [[M12|127]], [[M13|521]], [[M14|607]
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    ...hort hand used to refer to the 36th [[Mersenne prime]], specifically it is 2<sup>{{Num|2976221}}</sup>-1. This number was dicovered to be [[prime]] on 1 The corresponding [[perfect number]] 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
    |0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19||20||21||22||2 ...th prime 2 24737|24737]], [[Proth prime 2 55459|55459]], and [[Proth prime 2 67607|67607]] (current status [https://www.primegrid.com/stats_sob_llr.php
    5 KB (650 words) - 10:25, 26 March 2024
  • Paul Seelhoff
    :Found factor [[Proth prime 2 5|{{Kbn|+|5|2|39}}]] of {{DGF|36}}
    2 KB (195 words) - 00:13, 15 January 2024
  • Seventeen or Bust
    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 [ |format=,*[[%PAGE%|²{#titleparts:%TITLE%¦1¦2}²]]\n,,
    3 KB (544 words) - 16:44, 21 July 2019
  • M2
    | rank= 2 | pdigits= 2
    193 bytes (19 words) - 13:43, 17 February 2019
  • M3
    | digits= 2
    194 bytes (19 words) - 13:43, 17 February 2019
  • Perfect number
    ...r 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. ...irst four perfect 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}}. ...prime number, ([[Édouard Lucas]] having shown earlier that [[M12]], <math>2^{127}-1</math> is also prime), and it remained so until 1911. Prior to the
    2 KB (213 words) - 14:30, 17 February 2019
  • Double check
    ...volved 2 independent double checks. [[Mlucas]] and [[Glucas]] are used and 2 different processor types are used. [[Landon Curt Noll]]'s [[Mprime (Cray)|
    2 KB (373 words) - 15:08, 5 June 2019
  • Residue
    Here is the Lucas test for <math>2^7-1</math>, which is 127: :S1 = (4 * 4 - 2) mod 127 = 14
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    ! rowspan="2" | Hardware ...one, and the average GIMPS participant has completed less than three (mean 2.67).
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...variants) should work. (The [[Mfaktc#Future|next version]] will require CC 2.0 or newer.) **executables: CUDA >= 4.2 capable driver (295 series or newer)
    5 KB (765 words) - 14:54, 25 February 2019
  • Mfakto
    -v <n> verbosity level: 0=terse, 1=normal, 2=verbose, 3=debug -tf <exp> <min> <max> trial factor M<exp> from 2^<min> to 2^<max> and exit instead of parsing the worktodo file
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    *x<sub>2</sub> = f(x<sub>1</sub>) ...<sub>n+1</sub> = x<sub>n</sub><sup>2</sup> + a, where a <math>\neq </math>-2.
    3 KB (558 words) - 10:28, 6 February 2019
  • Riesel number
    ...ński problem]] article, [[Hans Riesel]] found in 1956 that [[Riesel prime 2 509203|{{Kbn|509203|n}}]] is always composite. *[[Riesel 2 Riesel|Riesel numbers]]
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  • M12
    ...son|Robinson]] at [[University of California, Los Angeles]] found [[M13]], 2<sup>521</sup>-1. ...mula (2<sup>2</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
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  • Primality test
    *[[Proth's theorem]]: Used to test numbers of the form {{Kbn|+|k|n}} with 2<sup>{{Vn}}</sup> > {{Vk}}, making it useful in several [[distributed comput
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  • Lucas primality test
    Prove that N = 811 is prime knowing that N-1 = 2 &times; 3<sup>4</sup> &times; 5 :<math>3^{810/2}\,= \,3^{405}\,\equiv \, 810\,\pmod{811}</math>
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  • Square root
    For example, <math>\sqrt 9 = 3</math> since <math>3^2 = 3 \times 3 = 9</math>. ...uadratic equations such as <math>x^2=9</math> or, more generally, <math>ax^2+bx+c=0</math>.
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Pépin's test
    ...r 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. ...<math>F_n = 2^{2^n}+1</math> is a prime if and only if <math>\ 3^{(F_n-1)/2} \ \equiv -1 \ \pmod{F_n}</math>.
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  • Proof by induction
    *'''Step 2''' :<math>\sum_{k=1}^{n}\,(2k-1)\,=\,n^{2}</math>
    4 KB (679 words) - 13:57, 20 February 2019
  • Floating-point
    ...and scaled using an [[exponent]]. The [[base]] for the scaling is normally 2, 10 or 16. The typical number that can be represented exactly is of the for
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  • Complex number
    ...on is <math>z = x + iy</math>. From the previous paragraph we get: <math>i^2 = -1</math>. ..._1 y_1 + x_2 y_2}{x_2^2 + y_2^2}\,+\,\frac {x_2 y_1 - x_1 y_2}{x_2^2 + y_2^2} \,i</math>
    2 KB (280 words) - 14:59, 26 March 2023
  • Operation Billion Digits
    *'''M(''exponent'') no factor from 2^(''startdepth'') to 2^(''end depth'')'''. ...system. They're currently at Level 21.07 (around twenty one candidates to 2^87). Their greatest factor was found at bit depth 85 by Åke Tilander, a fa
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  • M37
    ...hort hand used to refer to the 37th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|3021377}}</sup>-1. This number was discovered to be [[prime]] on
    877 bytes (111 words) - 11:04, 18 February 2019
  • Team Prime Rib
    ...ting power#P90 years|P90 CPU years]] of [[Lucas-Lehmer test|LL testing]] (#2 in [[PrimeNet]]) and over 7200 P90 CPU years of [[GIMPS factoring and sievi
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  • GIMPS factoring and sieving
    It is known that any factor of the Mersenne number <math>2^p-1</math> must be of the algebraic form <math>2kp+1</math> for some positi ...t an efficient way to do this, however! It is much easier to compute <math>2^p\,\bmod n</math>, i.e., the remainder after division by <math>n</math> by
    6 KB (962 words) - 10:08, 7 March 2019
  • Quadratic residue
    :<math>{x^2}\equiv{q}\ (mod\ p)</math>
    823 bytes (117 words) - 20:11, 26 October 2020
  • Modular square root
    :<math>r^2 \equiv a\ \pmod m</math> ...ulus. When this modulus is 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
    5 KB (726 words) - 10:38, 6 February 2019
  • Legendre symbol
    ...h; that is to say there exists an integer <math>k</math> such that <math>k^2 \equiv a \pmod{p}</math>, or in other words <math>a</math> is a quadratic r #<math>\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2} = \begin{cases} 1 & \text{if } p \equiv 1 \pmod{4} \\ -1 & \text{if } p \e
    2 KB (348 words) - 18:57, 28 September 2023
  • Law of quadratic reciprocity
    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>. *2 is a quadratic residue modulo <math>p</math> if and only if <math>p</math>
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  • Ten million digits
    In a period just over 2 weeks in summer 2008, the first two [[Mersenne prime]]s greater than {{Num|
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  • LLR
    ...gram]] available to perform primality test on numbers of the form {{Vk}}•2<sup>{{Vn}}</sup>±{{V|c}}. *the fastest algorithms are for base two numbers (with {{Vk}} < 2<sup>{{Vn}}</sup>):
    2 KB (300 words) - 22:00, 16 December 2023
  • Double Mersenne number
    ...d as '''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 *MM(2) = <math>2^3-1</math> = 7, known prime since antiquity
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  • P-1 factorization method
    :<math>E = 2^{E_2} * 3^{E_3} * 5^{E_5} * ... * B</math> where <math>E_2</math> is selected so that <math>2^{E_2}</math> is about B1 and the same for the other prime numbers. ''B'' is
    5 KB (814 words) - 01:35, 12 March 2019
  • Brent-Suyama extension
    A naive stage 2 would then compute T=S<sup>q</sup> = 3<sup>E*q</sup> for successive prime q ...s > 3 are of form 6k+/-1. Suppose instead that we compute T=S<sup>(6k)<sup>2</sup>-1</sup> = 3<sup>E*(6k-1)*(6k+1)</sup> whenever one of 6k+1 or 6k-1 is
    2 KB (421 words) - 11:51, 28 January 2019
  • Self-initializing quadratic sieve
    ...power. If somehow we find two integers X and Y such that <math>X^2\equiv Y^2\,\pmod N</math> and <math>X\not\equiv \pm Y\pmod N</math>, then <math>\gcd( ...values X and Y the method finds ''relations'' which have the form <math>t^2 \equiv u\,\pmod N</math> where u is the product of small [[prime]] numbers.
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  • Special number field sieve
    ...a\left(\exp\left( \left(\frac{32}{9}n\right)^{\frac{1}{3}} (\log n)^{\frac{2}{3}} \right)\right).</math>
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  • Miller-Rabin pseudoprimality test
    *If <math>m^2\equiv 1\,\pmod p</math> then <math>m\equiv 1\,\pmod p</math> or <math>m\equ ...<math>N</math> be an odd number being tested for primality, and <math>N = 2^n\,k + 1</math> where <math>k</math> is an odd number.
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  • Generalized Fermat number
    ...(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,2}</math> generates the [[Saouter number]]s.
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  • Patrick Laroche
    He is credited with discovery of the [[M51|51th known Mersenne prime]] 2<sup>{{Num|82589933}}</sup>-1.
    987 bytes (147 words) - 01:27, 15 January 2024
  • PrimeGrid
    ...Search|321 Prime Search]] searching for mega primes of the form {{Kbn|±|3|2|n}}. ...rch]] searching for primes of the forms {{Kbn|±|27|2|n}} and {{Kbn|±|121|2|n}}.
    3 KB (458 words) - 10:28, 26 March 2024
  • PrimeNet server
    ...software forced into service has had important benefits. Since one of the 2 new primes was 10 million decimal digits long, it qualified for one of the
    2 KB (381 words) - 14:05, 21 August 2019
  • 321 Search
    ...ect|distributed computing project]] to search for [[prime]]s of the form 3*2<sup>n</sup>-1.
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  • Sieve of Eratosthenes
    ...>2</math>. Since every second number after that will be divisible by <math>2</math>, we cross out every second number; all such numbers are composite. ...umber is <math>3</math>. We see that <math>3</math> is prime because <math>2 \geq \sqrt{3}</math>. Now we cross out every third number which hasn't been
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  • Top GIMPS teams
    | 2 || [[Curtis Cooper|UCM-curtisc]] || 26914623
    2 KB (206 words) - 09:56, 7 March 2019
  • Aliquot sequence
    :<math>s_0\ =\ 10,\ \sigma(10)=1\ +\ 2\ +\ 5\ +\ 10</math> :<math>s_1\ =\ 18\ -\ 10\ =\ 8,\ \sigma(8)\ =\ 1\ +\ 2\ +\ 4\ +\ 8</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>
    671 bytes (92 words) - 00:34, 30 January 2019
  • Prime
    ...1 that is only divisible by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. ...P) + 1. From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into Q, because each division would leave a remainder o
    2 KB (447 words) - 00:22, 10 July 2023
  • Twin prime
    | align="right" | 10<sup>1</sup> || align="right" | 2 | align="right" | 10<sup>2</sup> || align="right" | 8
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  • Proth prime 2 28831
    |Pb=2 |PCount=2
    581 bytes (64 words) - 19:18, 5 April 2023
  • Nomenclature and notation
    ...th>x</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 A number 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
    The work accomplished by one core of a hypothetical 1GHz Core 2 Duo [[CPU]] in one day. One P90 year equals 5.075 GHz-days. 1 TFLOPS equals
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  • Math rendering
    !colspan="2"| !colspan="2"|
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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/Factoring
    ...rime95]] and [http://linux.redbird.com/~alien88/sbfactor12.zip SBFactor v1.2]. Be warned that SBFactor doesn't allow resuming a factoring in the middle ...he sievedepth is around 49 and 50, the factorworth is best between 1.5 and 2, around 1.7. Save it and rename it to make_worktodo.bat. Run it. Then start
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  • Why participate in GIMPS?
    ...rfect numbers 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] ...g new. 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 ma
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  • Sieving
    ...+ b</math>, while in the second the sequence has the form <math>m\, =\, an^2 + bn + c</math>. If we need to find prime numbers, for all primes between 2 and the square root of the maximum member of the sequence, set to 1 those e
    3 KB (521 words) - 11:14, 6 February 2019
  • Smooth number
    :900 = 2<sup>2</sup> &times; 3<sup>2</sup> &times; 5<sup>2</sup>
    436 bytes (63 words) - 21:36, 3 February 2019
  • Sieving program
    *[[PPSieve]] (sieving 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 *[[AthGFNSieve]] (performing sieving of generalized Fermat numbers b<sup>2<sup>n</sup></sup>+1) http://www.underbakke.com/AthGFNsv/
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  • NewPGen
    *k&times;2<sup>n</sup>+1, k&times;2<sup>n+1</sup>+3 (Sophie Germain) ...sup>n</sup>+1, 2k&times;b<sup>n</sup>+1 (Cunningham Chain 2nd kind, length 2)
    3 KB (529 words) - 09:32, 7 March 2019
  • Sophie Germain prime
    ...prime. For example, 23 is a Sophie Germain prime because it is a prime and 2*23+1 = 47, also prime. | 1 || 2 || -
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  • M13
    '''M13''' is the 13th known [[Mersenne prime]] 2<sup>521</sup>-1 found on 1952-01-30 by [[Raphael M. Robinson]].
    419 bytes (48 words) - 14:50, 19 September 2021
  • 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]].
    430 bytes (49 words) - 14:49, 19 September 2021
  • M16
    '''M16''' is the 16th known [[Mersenne prime]] 2<sup>{{Num|2203}}</sup>-1 found on 1952-10-07 by [[Raphael M. Robinson]].
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  • M17
    '''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]].
    415 bytes (47 words) - 22:26, 17 February 2019
  • 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]].
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  • 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 an integer <math>a</math :<math>a^{(p-1)/2} \equiv -1\pmod{p}</math>.
    549 bytes (88 words) - 18:15, 28 September 2023
  • Proth prime
    ...is not a true class of numbers, but primes in the form {{Kbn|+|k|n}} with 2<sup>''n''</sup> > ''k'' are often called Proth primes.
    656 bytes (91 words) - 07:02, 31 August 2020
  • Riesel prime
    ...ion of a '''Riesel prime''' mostly all primes of the form {{Kbn|k|n}} with 2<sup>{{Vn}}</sup> > {{Vk}} are called like this on many pages. *[https://www.mersenneforum.org/showthread.php?t=8621 "RPS {{Vk}}*2^{{Vn}}-1, {{Vk}}<300 reservations/status"]: [https://www.mersenneforum.org/
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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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  • Floor function
    ! 2 | 2 || 2
    813 bytes (111 words) - 16:56, 29 August 2022
  • Power of two
    ...2 &times; 2 &times; 2 &times; ...). They are normally represented as <math>2^n</math> where n is the [[exponent]]. ...refix for multiples of 1024 is kibi-, so 1024 bytes = 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
    ...mple, 3 = 4 - 1 = 2<sup>2</sup> - 1 is a Mersenne prime; so is 7 = 8 - 1 = 2<sup>3</sup> - 1. See [[Mersenne prime]]. ...At the time of writing, a candidate at the current wavefront may take 1 to 2 weeks on a modern PC (say, a quad core Intel with AVX2 instructions or an A
    14 KB (2,370 words) - 15:15, 17 August 2019
  • 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
    ...of 1.92 or 2.06 then you know the correct value is 2. If you end up with 2.2 then the calculation is invalid. If you proved that the maximum accumulated ...of 1.92 or 2.06 then you know the correct value is 2. If you end up with 2.2 then the calculation is invalid. If you proved that the maximum accumulated
    12 KB (1,995 words) - 09:55, 7 March 2019
  • Overclocking
    ...GP PORT for your AGP card is set to operate at a fixed fraction, (2/3 or 1/2), of the FSB.
    14 KB (2,326 words) - 15:17, 11 February 2019
  • Area
    An '''area''' is defined as a surface enclosed by a shape. It can be 2 or 3 dimensional. For 3 dimensional shapes, the surface is the area.
    215 bytes (32 words) - 13:43, 18 September 2022
  • 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.)
    655 bytes (81 words) - 12:49, 25 March 2019
  • Keyboard shortcuts
    !rowspan="2"|{{Key|o}} !rowspan="2"|{{Key|e}}
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  • Multiple polynomial quadratic sieve
    ...two integers <math>X</math> and <math>Y</math> such that <math>X^2\equiv Y^2\,\pmod N</math> and <math>X\not\equiv \pm Y\pmod N</math>, then <math>\gcd( ...<math>Y</math> the method finds ''relations'' which have the form <math>t^2 \equiv u\,\pmod N</math> where <math>u</math> is the product of small [[pri
    6 KB (1,068 words) - 14:33, 13 February 2019
  • Proth number
    :{{V|N}} = {{Kbn|+|k|2|n}} ...is an odd positive [[integer]] and {{Vn}} is a positive integer such that 2<sup>{{Vn}}</sup> > {{Vk}}.
    670 bytes (104 words) - 10:59, 9 July 2021
  • 27121 Search
    ...r prime numbers of the form: 27 &times; 2<sup>n</sup> ± 1 and 121 &times; 2<sup>n</sup> ± 1. ...s]]. The bigger primes 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, f
    983 bytes (138 words) - 13:25, 8 February 2019
  • 12121 Search
    ...m: 121 &times; 2<sup>n</sup>-1, it also searches for primes of 121 &times; 2<sup>n</sup>+1 after being a partner of [[PrimeGrid]]. ...y 7 primes in the [[The Prime Pages]], their largest number is 121 &times; 2<sup>2033941</sup>-1 which weighs in at 612280 digits. This prime was found
    984 bytes (143 words) - 13:30, 8 February 2019
  • Assignments
    ...php?showsource=1] of FFT sizes and run times as calculated on the baseline 2.0 GHz Core2 Duo machine that set the current GHz-days credit standards. ...php?showsource=1] of FFT sizes and run times as calculated on the baseline 2.0 GHz Core2 Duo machine that set the current GHz-days credit standards (sam
    20 KB (3,473 words) - 18:42, 14 December 2023
  • GIMPS type of work
    *Standard (take the next exponent the server hands out, takes 1 to 2 weeks on typical hardware<ref>By typical, I am referring to a quad core Int ...ll yield a number with at least 100 million digits, these will take around 2 weeks on a Radeon VII, or several weeks on high end consumer hardware <ref>
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  • How to get started with GIMPS
    ...taticly linked ''sprimexxx.tar.gz'' for systems that do not have the glibc 2.1 runtime libraries. *'''OS/2''': Max Alekseyev has ported the Linux version to OS/2. Download his ''os2vxxx.zip'' zipped executable.
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  • GIMPS chances of finding a prime number
    ...ing as it slows down, and there are [[twin prime]]s - primes that are only 2 apart like 612,107 and 612,109 - and this makes matters more complicated. ...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 suppose
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  • Worktodo.txt
    *[[Elliptic curve method|ECM]]2=assignment ID,k,b,n,c,B1,B2,curves to do<nowiki>[</nowiki>,specific_sigma,B
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  • P+1 factorization method
    <math>\large U_0 = 0\,,\, U_1 = 1\,,\, V_0 = 2\,,\, V_1 = u </math> ...= uU_{n-1} - U_{n-2}\,,\, V_n = uV_{n-1} - V_{n-2}</math> for <math>n\geq 2</math>
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  • 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
    2. If you want to factor <math>a^{61}-1</math>, you can rewrite the power as ...culty 113) or <math>x^5-100</math> (difficulty 115) by the method of 1. or 2. alone.
    7 KB (1,238 words) - 16:14, 12 February 2019
  • NFSNET
    |<math>2^{772}+1</math> || 611380852128317600120825600011309662450676630495941840761 |<math>2^{779}+1</math> || 173158781290488639279749054806964483697237470930354987999
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  • Mersenneplustwo factorizations
    ...prime) and 2<sup>p</sup> is not divisible by 3 (it's only prime factor is 2). ...159965}}) found by ECMNet, as well as 36-digit factor of M<sub>11213</sub>+2 ({{FDBID|1000000000012161237}}) found using ''mprime'' (the linux version o
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  • ECMclient
    ...ized version using the [http://home.roadrunner.com/~mrodenkirch/ecmnet_3.0.2.zip source code]. 2. Move your homemade ecm.exe to the unzipped ecmclient folder (along with yo
    2 KB (383 words) - 11:16, 26 February 2019
  • GMP-ECM
    Distributed under GNU General Public License version 2. :*Get [http://mpir.org/mpir-2.5.1.tar.bz2 MPIR] and [https://gforge.inria.fr/frs/download.php/30965/ecm-6
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  • Gerbicz error checking
    ...]] to ensure validity of [[Proth prime|Proth]] tests and PRP tests on base-2 [[Riesel prime]] candidates, and by those programs and [[PRST]] in an exten # <math>u(t) \equiv (a^k)^{2^t} \pmod{p}</math>
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  • Jacobi error checking
    ...:Jacobi symbol|Jacobi symbol]] of the residue plus 2 <math>\left(\frac{Res+2}{M_p}\right)</math> has to be +1 *Jacobi symbol of the residue minus 2 <math>\left(\frac{Res-2}{M_p}\right)</math> has to be -1
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  • ECPP-DJ
    *[[BPSW test]] (strong PRP-2 test followed by extra strong Lucas-Selfridge test)
    2 KB (237 words) - 18:04, 2 December 2019
  • BLS75
    ...0025-5718-1975-0384673-1.pdf "New Primality Criteria and Factorizations of 2^{{V|m}} ± 1"]. ''Mathematics of Computation.'' Volume 29, Number 130: 620-
    584 bytes (85 words) - 20:12, 26 October 2020
  • Advanced Vector Extensions
    They add 512-bit vector operations capabilities in 2013, with up to 2 FMAs (Fused Multiply Add instructions), to accelerate performance of demand
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  • GIMPS PrimeNet reports
    ...eported. If the number has not had a successful [[double check]], the last 2 digits are masked. For unconfirmed results any error code is also displayed
    7 KB (1,137 words) - 15:30, 13 August 2020
  • M29
    | foundwith=[[Lucas-Lehmer test]] / [[NEC SX-2]]
    296 bytes (26 words) - 08:25, 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
    ...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]]. ...e, because it is multiple of both [[Mersenne number]]s 2<sup>p</sup>-1 and 2<sup>q</sup>-1.
    347 bytes (62 words) - 13:07, 19 February 2019
  • Square number
    ...of two square integers is also a square number (e.g. 2/3&nbsp;&times;&nbsp;2/3 = 4/9). :1<sup>2</sup> = 1
    3 KB (408 words) - 13:56, 19 February 2019
  • Square (algebra)
    Example: n squared equals n<sup>2</sup> = n &times; n.
    341 bytes (47 words) - 14:03, 19 February 2019
  • Editing
    == Level 2 == == Level 2 ==
    5 KB (805 words) - 06:50, 1 May 2019
  • AdvancedFactor
    ...{{Num|1000000}} and {{Num|2000000}} for factors between 2<sup>64</sup> and 2<sup>68</sup>. :bd is where to begin trial-factoring at (2<sup>bd</sup>).
    2 KB (300 words) - 16:08, 19 February 2019
  • Riesel problem 1
    {{DISPLAYTITLE:Riesel problem, {{Kbn|-|k|2|n}}, {{Vk}} < {{Num|509203}}}} ...wed that there are an infinite number of integers {{Vk}} such that {{Kbn|k|2|n}} is not prime for any integer {{Vn}}. He showed that the number {{Vk}} =
    6 KB (689 words) - 18:14, 4 April 2024
  • Tutorial LLR
    Sieved NewPGen-files are available for the projects 15k (phase 2), [http://mersenneforum.org/showthread.php?t=2667 PSP] and [http://www.mers ...ake sure the numbers 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
    ...889; it characterises the set (class, condition) of [[natural number]]s 1, 2, 3, etc., and consists of the following '''Peano postulates''' (also called :{{V|x}} + 2 = {{V|x}}<sup>+<sup>+</sup></sup>
    2 KB (269 words) - 17:04, 24 October 2020
  • Frequently suggested improvements
    ...ave a 0 [[residue]], we could start with that 0 residue and repeatedly add 2 and take a square root. Unfortunately, taking a modular square root is comp
    2 KB (392 words) - 14:37, 20 February 2019
  • LL test distribution
    ...Internet Mersenne Prime Search|GIMPS]]'s LL tests fall roughly into one of 2 categories. As time passes and GIMPS exhausts smaller exponents, both leadi ...on completion. The largest Mersenne number successfully double checked was 2<sup>{{Num|666666667}}</sup>-1, with {{Num|200686664}} digits. It was tested
    2 KB (415 words) - 21:32, 12 February 2020
  • ElevenSmooth
    ...ng 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
    ...abbreviated URL ''mersenne.ca/<numbers>'' where if the value is less than 2^32 it is assumed to be the exponent, otherwise it is assumed to be the fact ...Phase 2 of the factoring involves trial-factoring candidates to (at least) 2^64, using a simple [http://www.mersenne.ca/tf1G.php anonymous reservation s
    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://mat ::<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
    A '''Cullen number''' {{V|C<sub>n</sub>}} is a number of the form {{Kbn|+|n|2|n}}, ...}}, {{PP|89536|6679881}} and for no other {{Vn}} < {{Num|{{GP|Cullen prime 2|CuMaxn}}}} ({{OEIS|l|A005849}}).
    2 KB (252 words) - 17:39, 31 August 2021
  • Mtsieve
    |latest=2.4.5<br>2023-03-24 *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 4
    |CKMaxn={{#expr:floor({{GP|Carol-Kynea prime 2|CKMaxn}}/2)}} |CKDate={{GP|Carol-Kynea prime 2|CKDate}}
    389 bytes (47 words) - 10:31, 10 June 2019
  • Carol-Kynea prime 8
    |CKMaxn={{#expr:floor({{GP|Carol-Kynea prime 2|CKMaxn}}/3)}} |CKDate={{GP|Carol-Kynea prime 2|CKDate}}
    389 bytes (47 words) - 10:33, 10 June 2019
  • Carol-Kynea prime
    ...rs''' 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''' are [ ...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 forms
    8 KB (1,172 words) - 00:38, 6 July 2023
  • Multifactorial number
    :<math>n! = 1 \cdot 2 \cdot 3 \cdots (n{-}2) \cdot (n{-}1) \cdot n</math> :<math>n!! = (n) \cdot (n-2) \cdot (n-4) \cdots</math>
    560 bytes (81 words) - 14:36, 20 July 2021
  • Riesel prime 2 1
    |Rb=2 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
    |Rb=2 2;T:ST;C:{{NWo|+|2}}, {{NWi|MM|4|1}}
    5 KB (523 words) - 09:47, 5 October 2023
  • Riesel prime 2 5
    |Rb=2 2
    2 KB (196 words) - 18:36, 9 October 2021
  • Riesel prime 2 7
    |Rb=2
    3 KB (264 words) - 22:08, 5 July 2023
  • Riesel 2 Low-weight
    category=Riesel 2 Low-weight [[Category:Riesel 2 Low-weight| ]]
    948 bytes (121 words) - 13:08, 21 July 2021
  • Riesel prime 2 15
    |Rb=2 2;T:T
    2 KB (252 words) - 13:29, 5 May 2024
  • Riesel prime 2 2145
    |Rb=2
    3 KB (329 words) - 07:59, 17 August 2021
  • Riesel prime 2 2805
    |Rb=2
    2 KB (185 words) - 04:16, 5 May 2024
  • Riesel prime 2 10391
    |Rb=2 |RCount=2
    236 bytes (24 words) - 00:24, 17 July 2021
  • Carol-Kynea prime 16
    |CKMaxn={{#expr:floor({{GP|Carol-Kynea prime 2|CKMaxn}}/4)}} |CKDate={{GP|Carol-Kynea prime 2|CKDate}}
    390 bytes (47 words) - 10:34, 10 June 2019
  • Williams prime PM 2
    |WiBase=2 |WiMaxn={{GP|Riesel prime 2 3|RMaxn}}
    246 bytes (35 words) - 14:04, 1 August 2021
  • Williams prime MP 2
    |WiBase=2 |WiMaxn={{GP|Proth prime 2 1|PMaxn}}
    243 bytes (35 words) - 08:06, 1 August 2021
  • Williams prime PP 2
    |WiBase=2 |WiMaxn={{GP|Proth prime 2 3|PMaxn}}
    259 bytes (36 words) - 10:52, 13 July 2021
  • Williams prime MM 4
    |WiMaxn={{#expr:floor({{GP|Riesel prime 2 3|RMaxn}}/2)}} |WiDate={{GP|Riesel prime 2 3|RDate}}
    265 bytes (37 words) - 20:50, 31 July 2021
  • Williams prime MM 3
    |WiMaxn={{GP|Riesel prime 3 2|RMaxn}} |WiDate={{GP|Riesel prime 3 2|RDate}}
    248 bytes (35 words) - 09:56, 16 March 2023
  • Williams prime MM 2
    |WiBase=2 |WiMaxn={{GP|Riesel prime 2 1|RMaxn}}
    244 bytes (35 words) - 20:47, 31 July 2021
  • Williams prime MM 6
    2
    642 bytes (56 words) - 09:01, 22 November 2023
  • Williams prime MM 7
    2
    430 bytes (42 words) - 11:00, 21 May 2019
  • Williams prime MM 8
    |WiMaxn={{#expr:floor({{GP|Riesel prime 2 7|RMaxn}}/3)}} |WiDate={{GP|Riesel prime 2 7|RDate}}
    291 bytes (42 words) - 20:52, 31 July 2021
  • Williams prime MM 9
    2
    412 bytes (45 words) - 20:59, 31 July 2021
  • Williams prime MM 12
    2
    318 bytes (29 words) - 11:41, 21 May 2019
  • Williams prime MM 13
    2
    394 bytes (38 words) - 20:54, 31 July 2021
  • Williams prime MM 16
    |WiMaxn={{#expr:floor({{GP|Riesel prime 2 15|RMaxn}}/4)}} |WiDate={{GP|Riesel prime 2 15|RDate}}
    270 bytes (37 words) - 20:56, 31 July 2021
  • Williams prime MM 18
    2
    218 bytes (19 words) - 12:22, 21 May 2019
  • Williams prime MM 21
    2
    231 bytes (19 words) - 12:53, 21 May 2019
  • Williams prime MM 22
    2
    221 bytes (19 words) - 13:00, 21 May 2019
  • Riesel prime 2 31
    |Rb=2
    2 KB (228 words) - 19:30, 19 July 2023
  • Williams prime MM 30
    2
    233 bytes (19 words) - 13:24, 21 May 2019
  • Williams prime MM 32
    |WiMaxn={{#expr:floor({{GP|Riesel prime 2 31|RMaxn}}/5)}} |WiDate={{GP|Riesel prime 2 31|RDate}}
    270 bytes (37 words) - 21:00, 31 July 2021

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