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  • Williams prime
    ...-> [[Williams prime MM 4|page]] || {{OEIS|A326655}} -> [[Williams prime MP 4|page]] || ||
    5 KB (744 words) - 07:30, 5 August 2019
  • Carol-Kynea prime 2
    4
    5 KB (537 words) - 08:17, 9 October 2020
  • Carol-Kynea prime 6
    4
    1 KB (85 words) - 10:45, 16 April 2023
  • Splitting a sieve file
    Split the sieve file <code>t16_b999.prp</code> into 4 separate files. *Call <code>gawk -f split.awk files=4 t16_b999.prp</code>.
    1 KB (203 words) - 18:52, 2 October 2022
  • 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
  • Mersenne prime
    ...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 5. ...f {{V|M<sub>p</sub>}} evenly divides <math>S_{p-2}</math>, where <math>S_0=4</math> and for <math>k>0</math>, <math>S_k=S_{k-1}^2-2</math>.
    5 KB (857 words) - 14:53, 19 September 2021
  • Mersenne number
    *Let <math>q = 3 \pmod{4}</math> be a prime. <math>2q+1</math> is also a prime if and only if <math>
    2 KB (351 words) - 11:28, 7 March 2019
  • Marin Mersenne
    : <math>\sqrt[4]{\frac{2}{3-\sqrt{2}}}</math>
    11 KB (1,582 words) - 01:17, 15 January 2024
  • George Woltman
    | MersForum=4 Prime95
    1 KB (164 words) - 14:40, 21 August 2019
  • Fermat number
    :{{V|F}}<sub>2</sub> = {{Kbn|+|4}} = 17 :{{V|F}}<sub>4</sub> = {{Kbn|+|16}} = 65537
    12 KB (1,913 words) - 14:35, 9 August 2021
  • Derrick Henry Lehmer
    ...math> dealing with two cases of the [[Ramanujan conjecture]], that <math>5^4 | p(599)</math> and <math>11^3 | p(721)</math>, computing the values using
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Cunningham project
    Only the first five Fermat numbers, k = 0, 1, 2, 3, 4 corresponding to 3, 5, 17, 257 and 65537 are known to be prime. All Fermat
    7 KB (1,150 words) - 23:48, 19 April 2023
  • Lucas-Lehmer test
    ...991) 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 from t ...he smallest unit of computer memory). The value of S<sub>1</sub> has about 4 (= 2<sup>2</sup>) bits, the value of S<sub>2</sub> has about 8 (= 2<sup>3</
    20 KB (3,572 words) - 14:30, 17 February 2019
  • Édouard Lucas
    '''François Édouard Anatole Lucas''' {{BirthDeath|4. April 1842|3. October 1891}} was born in Amiens, France and educated at th
    2 KB (296 words) - 01:09, 15 January 2024
  • Decimal
    ...bols (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. These dig
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    ...16) is used because every 4 digits can be used as short for binary. Every 4 binary digits turns into one hexadecimal digit when changing between them.
    2 KB (399 words) - 10:37, 18 January 2019
  • Addition
    :2 + 2 = 4
    333 bytes (43 words) - 16:55, 29 August 2022
  • Exponent
    ...a &times; a &times; a &times; a is written as <math>a^4</math>, the number 4 is the index, or exponent.
    1 KB (273 words) - 16:56, 29 August 2022
  • Factorial number
    :5! = 5 * 4 * 3 * 2 * 1 = 120 :10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3628800
    729 bytes (93 words) - 13:40, 5 November 2023
  • Composite number
    ...te''' if it is neither [[prime]] nor equal to 1. The smallest composite is 4.
    358 bytes (56 words) - 23:30, 26 October 2020
  • Subtraction
    ...ion are, minuend &minus; subtrahend = difference. The expression 7 &minus; 4 = 3 can be spoken as "seven minus four equals three", "seven take away four
    893 bytes (128 words) - 16:58, 29 August 2022
  • M47
    **One on 4 quad-core SPARC64 VII 2.52GHz CPUs of a Sun SPARC Enterprise M8000 Server
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    |style="text-align:left"|Intel Pentium 4||3078||1||72.40||162.02||14.91||86 |style="text-align:left"|AMD Phenom II X4||3414||4||34.86||76.27||4.59||125
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Even number
    All numbers ending in 0, 2, 4, 6, or 8 are even.
    425 bytes (61 words) - 11:19, 7 March 2019
  • M41
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2.4 GHz Pentium 4 [[Personal computer|PC]] The discovery took 14 days of computing on 2.4 GHz Pentium 4 Windows XP PC.
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2 GHz Pentium 4 [[Personal computer|PC]]
    1 KB (191 words) - 11:31, 18 February 2019
  • Riesel prime 2 21143
    4
    403 bytes (26 words) - 18:40, 2 January 2023
  • Proth prime 2 12321
    4
    498 bytes (31 words) - 13:34, 2 January 2023
  • Program
    ...Programming, Volume 1, 3rd Edition, 1997, Addison-Wesley, ISBN 0-201-89683-4
    2 KB (263 words) - 11:53, 7 February 2019
  • M42
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 2.4 GHz Pentium 4 [[Personal computer|PC]]
    934 bytes (118 words) - 11:26, 18 February 2019
  • Fast Fourier transform
    ...he history of the fast Fourier transform," ''IEEE ASSP Magazine'' '''1''' (4), 14-21 (1984).
    17 KB (2,684 words) - 18:50, 28 September 2023
  • Primo
    |latest=4.3.2<br><small>2020-02-06</small> ...ood]] with Primo 4.3.0. The certification process took 21.5 months using a 4 x 12 core AMD 6174.
    1 KB (191 words) - 20:33, 12 May 2020
  • M32
    :"Took 26.562767 minutes to calculate using Maple 4.0 on a 512-MW 4 CPU Cray 2"
    2 KB (279 words) - 08:35, 18 February 2019
  • 2 Plus Tables
    *4 : {{FDBID|17}}
    2 KB (127 words) - 15:28, 17 August 2019
  • Binary
    :1 &rarr; 1 * 2<sup>4</sup> = 16
    1 KB (210 words) - 11:16, 22 January 2019
  • Parallel computing
    | 4 || 7200 || + || 86400 || 93600 ||
    3 KB (416 words) - 06:47, 1 May 2019
  • M40
    The discovery took 19 days of computation on a 2 GHz Pentium 4 Dell Dimension PC at a Michigan State University lab.
    1 KB (189 words) - 11:17, 18 February 2019
  • M44
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 3 GHz Pentium 4 [[Personal computer|PC]]
    997 bytes (129 words) - 11:35, 18 February 2019
  • Modular arithmetic
    ...3 hours = 2 PM). We can also subtract: 2-3 = 11 and even multiply: 5&times;4 = 8 (because 5+5+5+5 = 8). This is arithmetic modulo 12 and the set of numb
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    :We get: a' = b' = 3 &times; 8 mod 5 = 4 :x = a' b' = 4 &times; 4 = 16
    4 KB (582 words) - 17:01, 29 August 2022
  • Elliptic curve method
    ...''' = '''S'''. Given a point '''P''', we could compute 2'''P''', 3'''P''', 4'''P''', and so on. For some value <math>g</math>. we will have <math>g</mat where: <math>S = \frac {A+2}4</math> which can be precomputed when the elliptic curve is selected and
    19 KB (3,181 words) - 22:27, 6 July 2023
  • University of California, Los Angeles
    ...ty. [[University of Central Missouri]] has contributed to the discovery of 4, making it second to UCLA.
    2 KB (347 words) - 14:54, 19 September 2021
  • Raphael M. Robinson
    ...universal Turing machine," describing a universal [[Turing machine]] with 4 symbols and 7 states;
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    | top5000id=4
    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||23||24|
    5 KB (650 words) - 10:25, 26 March 2024
  • M4
    | rank= 4 | pdigits= 4
    195 bytes (19 words) - 13:44, 17 February 2019
  • M5
    | digits= 4
    204 bytes (18 words) - 13:46, 17 February 2019
  • Perfect number
    ...sitive 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. :for ''n'' = 5: &nbsp; 2<sup>4</sup>(2<sup>5</sup> - 1) = 496
    6 KB (885 words) - 11:33, 7 March 2019
  • Residue
    ...roneaous residues (meaning they both missed a prime) out of a pool of ~ 18.4 pentillion numbers, this is considered to be impossible. :S0 = 4
    1 KB (235 words) - 10:24, 6 February 2019
  • Mfakto
    ...this will start happening at vector size 8. On a HD6870, a vector size of 4 is fastest except for the barrett92 and barrett72 kernels which run slightl Allowed sizes are 1, 2, 4, 8, 16.
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    ...rating the polynomial, and at the same time computing x<sub>2</sub>, x<sub>4</sub>, x<sub>6</sub> and so on by iterating the polynomial twice. Then we c ...o compute gcd(x<sub>1</sub> - x<sub>2</sub>, N), gcd(x<sub>2</sub> - x<sub>4</sub>, N), gcd(x<sub>3</sub> - x<sub>6</sub>, N), and so on until we find a
    3 KB (558 words) - 10:28, 6 February 2019
  • M12
    ...} [http://www.garlic.com/~wedgingt/MMPstats.txt], a [[bit level]] over 169.4. The current version of [[Prime95]] cannot handle numbers this large, nor c
    2 KB (354 words) - 14:52, 19 September 2021
  • Lucas primality test
    Prove that N = 811 is prime knowing that N-1 = 2 &times; 3<sup>4</sup> &times; 5
    1 KB (177 words) - 14:31, 17 February 2019
  • Square root
    ...= 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + ...</math> <u> 1 2. 3 4 </u>
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Pépin's test
    ...of other numbers, like the [[Generalized Fermat number]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1</math> with k = 5 instead of k = 3.
    2 KB (401 words) - 14:40, 6 March 2019
  • Proof by induction
    ...ally taken to be 1, but that is not essential. In some proofs (see example 4 below) we have to show that the statement is true for several values of n. :<math>\sum_{k=1}^{n}k^{3}\,=\,\frac{n^{2}(n+1)^{2}}{4}</math>
    4 KB (679 words) - 13:57, 20 February 2019
  • Pseudoprime
    ...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>.
    1 KB (155 words) - 20:32, 25 July 2020
  • Modular square root
    ===Modulus congruent to 3 modulo 4=== :<math>r\equiv \pm a^{(m+1)/4}\ \pmod m</math>
    5 KB (726 words) - 10:38, 6 February 2019
  • Legendre symbol
    ...es} 1 & \text{if } p \equiv 1 \pmod{4} \\ -1 & \text{if } p \equiv 3 \pmod{4} \end{cases}</math> ...property is known as the [[law of quadratic reciprocity]]. The properties 4 and 5 are traditionally known as the ''supplements'' to quadratic reciproci
    2 KB (348 words) - 18:57, 28 September 2023
  • 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 only if *If both of <math>p</math> or <math>q</math> are congruent to 3 mod 4: <math>p</math> is a quadratic residue modulo <math>q</math> if and only if
    1 KB (208 words) - 18:19, 2 October 2022
  • Double Mersenne number
    This holds true for the first 4 terms: However this does not hold true for next 4 terms:
    4 KB (655 words) - 14:50, 19 September 2021
  • Self-initializing quadratic sieve
    :<math>45^2\,\equiv \,2^4*7^0*13^1</math>
    10 KB (1,763 words) - 02:56, 12 March 2019
  • Miller-Rabin pseudoprimality test
    ...strated that a composite <math>N</math> will only pass at most <math>(N-1)/4</math> such tests. ...35. Since the exponent is 4, the sequence will use exponents from zero to 4.
    3 KB (432 words) - 15:33, 28 January 2019
  • Generalized Fermat number
    3*2^41+1 is a Factor of xGF(38,4,3)!!!! (0.000000 seconds)
    5 KB (726 words) - 09:57, 12 September 2021
  • Patrick Laroche
    ...hunting on his media server to "give back" to the project. After less than 4 months and on just his fourth try, he discovered the new prime number. By w
    987 bytes (147 words) - 01:27, 15 January 2024
  • Top GIMPS teams
    | 4 || [[Team Prime Rib|Ars Technica Team Prime Rib]] || 4457877
    2 KB (206 words) - 09:56, 7 March 2019
  • Aliquot sequence
    :<math>s_1\ =\ 18\ -\ 10\ =\ 8,\ \sigma(8)\ =\ 1\ +\ 2\ +\ 4\ +\ 8</math> :<math>2^4\ *\ 31</math>
    6 KB (914 words) - 19:49, 21 February 2023
  • 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
    ...whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 x 5 ... x P) + 1. From the form of the number Q, it is obvious that no inte :the remainder r can be only 0, 1, 2, 3, 4, or 5
    2 KB (447 words) - 00:22, 10 July 2023
  • Twin prime
    | align="right" | 10<sup>4</sup> || align="right" | 205
    2 KB (255 words) - 06:08, 21 February 2023
  • Proth prime 2 28831
    4
    581 bytes (64 words) - 19:18, 5 April 2023
  • Math rendering
    | <code>\sideset{_1^2}{_3^4}\prod_a^b</code> || <math>\sideset{_1^2}{_3^4}\prod_a^b</math> | <code>{}_1^2\!\Omega_3^4</code> || <math>{}_1^2\!\Omega_3^4</math>
    33 KB (4,920 words) - 10:54, 7 March 2019
  • Why participate in GIMPS?
    ...d 28 who are equal to the sum of their aliquot divisors: 6 = 1+2+3, 28=1+2+4+7+14). He realized that the even perfect numbers (no odd perfect numbers ar
    7 KB (1,252 words) - 09:47, 7 March 2019
  • Sophie Germain prime
    | 4 || 11 || {{NRi|3|2}} | 17 || 239 || {{NRi|15|4}}
    1 KB (171 words) - 04:26, 3 November 2020
  • Torture test
    ...sing the default "Blend" setting, the first instance each will use about 3/4 of memory the system memory. The second instance will try to do the same. T
    2 KB (323 words) - 10:59, 18 June 2019
  • Introductory stuff
    ...[prime]] number that is one less than a [[power of two]]. For example, 3 = 4 - 1 = 2<sup>2</sup> - 1 is a Mersenne prime; so is 7 = 8 - 1 = 2<sup>3</sup
    14 KB (2,370 words) - 15:15, 17 August 2019
  • 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
  • Square (geometry)
    A '''square''' is a regular polygon with 4 sides of equal length and with 4 equal interior angles (90 degrees), thus having opposing sides parallel. A square is the most constrained of all types of 4 sided figures.
    296 bytes (42 words) - 13:44, 18 September 2022
  • Pocklington's theorem
    ...the [[KonyaginPomerance|Konyagin-Pomerance Test]] can be used. If <math>f^4 > n</math>, there is a polynomial-time algorithm, due to Coppersmith and Ho ...simplest one which will work for the value of ''f'' you have. For <math>f^4 < n</math>, there is no known algorithm which enhances Pocklington's theore
    2 KB (346 words) - 19:51, 30 August 2019
  • Multiple polynomial quadratic sieve
    :<math>45^2\,\equiv \,2^4*7^0*13^1</math>
    6 KB (1,068 words) - 14:33, 13 February 2019
  • How to get started with GIMPS
    ==Step 4==
    4 KB (623 words) - 13:39, 26 March 2019
  • GIMPS networking, PrimeNet and communication
    Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),
    8 KB (1,269 words) - 10:09, 7 March 2019
  • Strong law of small numbers
    ...numbers. [[Double Mersenne number]]s are a good example of this. The first 4 terms yield primes, but factors for next four terms have been found (after
    1 KB (197 words) - 15:02, 11 February 2019
  • P+1 factorization method
    If we define <math>D = u^2 - 4</math>, then for any odd prime <math>p</math>, <math>p</math> divides both
    8 KB (1,536 words) - 11:35, 12 February 2019
  • SNFS polynomial selection
    ...lo ''N'', the number to be factored. Typically one polynomial is of degree 4, 5 or 6 (the algebraic side) and the other one is linear (the rational side 4. For <math>a^{11k}-1</math> and <math>a^{13k}-1</math> you'd get degrees 10
    7 KB (1,238 words) - 16:14, 12 February 2019
  • ECMclient
    4. Run ecmclient.exe and it should get some work and start working on it.
    2 KB (383 words) - 11:16, 26 February 2019
  • GMP-ECM
    ....1.tar.bz2 MPIR] and [https://gforge.inria.fr/frs/download.php/30965/ecm-6.4.3.tar.gz GMP-ECM] (don't unpack them, just place them on your desktop). 4. Time to open MSYS again. Inside MSYS you'll need to do as shown on this sc
    4 KB (567 words) - 10:54, 6 December 2019
  • Gerbicz error checking
    *in [[Prime95]] since version 29.4 ...ot hold, then we roll back to the last stored <math>u</math> value (value (4) above). If we roll back too much (e.g. 100 times to the same term), then w
    3 KB (528 words) - 14:59, 3 October 2023
  • History
    ...ng to [[Odd Magnar Strindmo]]; but the discovery goes unnoticed until June 4.
    3 KB (479 words) - 10:55, 7 March 2019
  • System requirements and compatible hardware
    :Windows NT (3 / 4 / 4 SP 1 / 4 SP 2 / 4 SP 3 / 4 SP 4 / 4 SP 5 / 4 SP 6) :Windows 2000 (SP 1 / SP 2 / SP 3 / SP 4)
    766 bytes (95 words) - 12:25, 19 February 2019
  • PrimeNet summary report
    ...[exponent]]s of the [[Mersenne number]]s by ranges. It is broken down into 4 sets of vertical columns. The exponents below {{Num|10000000}} are treated
    7 KB (1,072 words) - 13:10, 19 February 2019
  • Square number
    ...square integers is also a square number (e.g. 2/3&nbsp;&times;&nbsp;2/3 = 4/9). :2<sup>2</sup> = 4
    3 KB (408 words) - 13:56, 19 February 2019
  • Editing
    ==== Level 4 ==== ==== Level 4 ====
    5 KB (805 words) - 06:50, 1 May 2019
  • Riesel problem 1
    | 4 || 7 || {{Num|155274}} || {{Num|{{PAGESINCATEGORY:Riesel 2 1Intervals2|page ! scope="row" | [[:Category:Riesel 2 1Intervals4|4]]
    6 KB (689 words) - 18:14, 4 April 2024
  • Mersenne.ca
    ...most efficient type of work for their particular CPU. Multiple CPUs (up to 4) can be compared directly to show relative efficiency comparison on all wor
    9 KB (1,396 words) - 15:42, 25 February 2019
  • Aurifeuillian factor
    **<math>t\equiv 1 \pmod 4</math> and <math>n\equiv t \pmod{2t}</math> **<math>t\equiv 2, 3 \pmod 4</math> and <math>n\equiv 2t \pmod{4t}</math>
    10 KB (1,257 words) - 08:04, 24 June 2019
  • Carol-Kynea prime 4
    |CKBase=4
    389 bytes (47 words) - 10:31, 10 June 2019
  • Carol-Kynea prime
    ...integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where <math>n \bmod 2 \equ
    8 KB (1,172 words) - 00:38, 6 July 2023
  • Multifactorial number
    :<math>n!! = (n) \cdot (n-2) \cdot (n-4) \cdots</math>
    560 bytes (81 words) - 14:36, 20 July 2021
  • Riesel prime 2 1
    2;T:ST;C:'''[[M1]]''', {{NWo|+|1}}, {{NWo|-|2}}, {{NWo|4|1}} 5;C:'''[[M3]]''', {{NWo|+|3}}, {{NWo|4|2}}, {{NWo|32|1}}
    2 KB (288 words) - 11:41, 3 April 2023

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