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  • Williams prime MM 5
    3
    917 bytes (86 words) - 12:13, 22 May 2019
  • Williams prime
    ...ime MM table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime MM|pages}}-3}} bases || [[Williams prime MM least|here]]<ref>The list contains values fo ...ime MP table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime MP|pages}}-3}} bases || [[Williams prime MP least|here]]<br>{{PAGESINCATEGORY:Williams p
    5 KB (744 words) - 07:30, 5 August 2019
  • Carol-Kynea prime 2
    3 3
    5 KB (537 words) - 08:17, 9 October 2020
  • Carol-Kynea prime 6
    3
    1 KB (85 words) - 10:45, 16 April 2023
  • Examples
    <pre><math>\sum_{i=1}^\infty \frac{1}{p_i} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \dotsb = \infty</math></pre> :<math>\sum_{i=1}^\infty \frac{1}{p_i} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \dotsb = \infty</math>
    11 KB (1,236 words) - 14:41, 3 September 2020
  • Mersenne prime
    ...us; 1 = {{Kbn|4}}, for example, is not a prime, because 15 is divisible by 3 and 5. The first four Mersenne primes {{V|M<sub>2</sub>}}, {{V|M<sub>3</sub>}}, {{V|M<sub>5</sub>}}, {{V|M<sub>7</sub>}} were known in antiquity.
    5 KB (857 words) - 14:53, 19 September 2021
  • Mersenne number
    ...for ''d=0 ... 48'' and has provided a new (and clearer) proof for case ''d=3''. *Let <math>q = 3 \pmod{4}</math> be a prime. <math>2q+1</math> is also a prime if and only i
    2 KB (351 words) - 11:28, 7 March 2019
  • Marin Mersenne
    : <math>\sqrt[4]{\frac{2}{3-\sqrt{2}}}</math> ...enne prime|Mersenne primes]] less than or equal to M(257) were:<br>M(2), M(3), M(5), M(7), M(13), M(17), M(19), M(31), ''M(67)'', [[M12|M(127)]], ''M(25
    11 KB (1,582 words) - 01:17, 15 January 2024
  • Fermat number
    :{{V|F}}<sub>0</sub> = {{Kbn|+|1}} = 3 :{{V|F}}<sub>3</sub> = {{Kbn|+|8}} = 257
    12 KB (1,913 words) - 14:35, 9 August 2021
  • Derrick Henry Lehmer
    ...the [[Ramanujan conjecture]], that <math>5^4 | p(599)</math> and <math>11^3 | p(721)</math>, computing the values using the [[Hardy-Ramanujan series]]
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Home prime
    The length of such chains is also of interest, in this case #HP(25) = 3. :21<sub>10</sub> = 3 &times; 7 (11 &times; 111)
    980 bytes (143 words) - 13:22, 6 March 2019
  • Cunningham tables
    *[[3 Minus Tables]] *[[3 Plus Tables]]
    614 bytes (69 words) - 12:08, 13 February 2019
  • Cunningham project
    ...ion]] 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 selecte | 2 || 3 || 5 || 6 || 7 || 10 || 11 || 12
    7 KB (1,150 words) - 23:48, 19 April 2023
  • 2 Minus Tables
    *2 : {{FDBID|3}} / M1 *3 : {{FDBID|7}} / M2
    2 KB (176 words) - 12:01, 13 February 2019
  • Lucas-Lehmer test
    ...<sup>2</sup> - 2 = 37634. So the fact that 2<sup>5</sup> - 1 divides S<sub>3</sub> (37634 / 31 = 1214) shows that 2<sup>5</sup> - 1 is prime. ...4 (= 2<sup>2</sup>) bits, the value of S<sub>2</sub> has about 8 (= 2<sup>3</sup>) bits, and the value of S<sub>50000000</sub> (which would need to be
    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 the Ecole Normal
    2 KB (296 words) - 01:09, 15 January 2024
  • Digit
    A '''digit''' is a symbol (a number symbol, e.g. "3" or "7") used in numerals (combinations of symbols, e.g. "37"), to represen
    1 KB (171 words) - 10:17, 18 January 2019
  • Decimal
    ...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. These
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    | 3 || '''11''' || 3
    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
    ...umber]]s (1, 2, 3, &hellip;), their negatives (&minus;1, &minus;2, &minus;3, ...) and the number zero. The set of all integers is usually denoted in [[
    3 KB (404 words) - 14:58, 26 March 2023
  • 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
  • Subtraction
    ...are, minuend &minus; subtrahend = difference. The expression 7 &minus; 4 = 3 can be spoken as "seven minus four equals three", "seven take away four lea
    893 bytes (128 words) - 16:58, 29 August 2022
  • Division
    :<math>2 \times 3 = 6</math>.
    2 KB (368 words) - 16:58, 29 August 2022
  • Remainder
    ...ng. 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
    ...factors (half of the numbers are multiples of 2, a third are multiples of 3 and so on) it pays to run factorization methods in the first category and t
    4 KB (642 words) - 12:57, 5 March 2019
  • 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
  • M46
    | foundwith=[[Lucas-Lehmer test]] / [[Prime95]] on 3 GHz Core 2 [[Personal computer|PC]] ...since 1996 testing over 1400 candidates. The calculation took 29 days on a 3.0 GHz Intel Core2 processor.
    2 KB (248 words) - 11:45, 18 February 2019
  • Riesel prime 2 187605
    3
    490 bytes (35 words) - 12:22, 11 December 2022
  • Program
    *Knuth, Donald E., The Art of Computer Programming, Volume 3, 3rd Edition, 1997, Addison-Wesley, ISBN 0-201-89685-0
    2 KB (263 words) - 11:53, 7 February 2019
  • M1
    | number= 3
    193 bytes (19 words) - 13:43, 17 February 2019
  • Conjectures 'R Us
    ==Sub-project #3==
    3 KB (503 words) - 02:20, 1 May 2024
  • Fast Fourier transform
    ...key algorithm is O(&epsilon; log ''n''), compared to O(&epsilon; ''n''<sup>3/2</sup>) for the naive DFT formula (Gentleman and Sande, 1966), where &epsi ...achlass: Theoria interpolationis methodo nova tractata," ''Werke'' band '''3''', 265&ndash;327 (K&ouml;nigliche Gesellschaft der Wissenschaften, G&ouml;
    17 KB (2,684 words) - 18:50, 28 September 2023
  • GpuOwL
    Apart from the LL test, gpuOwL also implements a base-3 [[probable prime|PRP]] test with [[Gerbicz error checking]] ensuring correc
    1 KB (216 words) - 05:22, 1 December 2020
  • M32
    ...nski was notified and tested the number on a 16 CPU Cray-C90, taking about 3 hours. Later it was verified on a different computer architecture, using di
    2 KB (279 words) - 08:35, 18 February 2019
  • 2 Plus Tables
    *1 : {{FDBID|3}}
    2 KB (127 words) - 15:28, 17 August 2019
  • Binary
    :0 &rarr; 0 * 2<sup>3</sup> = 0
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...) 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).
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    | 3 || 300 || x || 1440 || 432000 || ..., they could all be done in parallel. This would cut a 5 step procedure to 3. If the numbers were each 100 digits long and 10 individuals (or cores in a
    3 KB (416 words) - 06:47, 1 May 2019
  • Trial factoring
    ...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 N would b ...ce 2*7 is excluded because 2 will have been tried, 3*7 is excluded because 3 will have been tried, and 5*7 is excluded because 5 will have been tried. T
    7 KB (1,221 words) - 13:20, 11 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
  • Odd Magnar Strindmo
    Strindmo's 3 GHz Core 2 Duo PC first reported the prime to GIMPS on 2009-04-12. However,
    991 bytes (141 words) - 00:33, 15 January 2024
  • Modular arithmetic
    ...rithmetic 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>. The set <b>Z</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
    :Let m=5 , n=3 , a = b = 3. :We get: a' = b' = 3 &times; 8 mod 5 = 4
    4 KB (582 words) - 17:01, 29 August 2022
  • Elliptic curve method
    ...cation|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 only t ...for ''y''. If we plug in ''x'' = 2, we get out ''y'' = 23. Plug in ''x'' = 3, we get out ''y'' = 48. If we plug in enough different values for ''x'' and
    19 KB (3,181 words) - 22:27, 6 July 2023
  • Raphael M. Robinson
    ...ey.edu/sites/default/files/imagecache/photo_album_main/photo_albums/ROBINS~3.jpg ...[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
  • Sierpiński problem
    *{{Kbn|+|78557|2n}} is multiple of 3. *{{Kbn|+|78557|9n+3}} is multiple of 73.
    5 KB (650 words) - 10:25, 26 March 2024
  • Paul Seelhoff
    :28 [[Proth prime]]s for 3 &le; {{Vk}} &le; 97
    2 KB (195 words) - 00:13, 15 January 2024
  • M2
    | nvalue= 3
    193 bytes (19 words) - 13:43, 17 February 2019
  • M3
    | rank= 3 | pdigits= 3
    194 bytes (19 words) - 13:43, 17 February 2019
  • M4
    | digits= 3
    195 bytes (19 words) - 13:44, 17 February 2019
  • Perfect number
    ...ct number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect n :for ''n'' = 3: &nbsp; 2<sup>2</sup>(2<sup>3</sup> - 1) = 28
    6 KB (885 words) - 11:33, 7 March 2019
  • Unused computing power
    ..."do nothing". A [[computer]] that is a quad-[[Processor|core]] running at 3 GHz is capable of performing 12 billion instructions a second. Often less t
    971 bytes (156 words) - 13:46, 25 January 2019
  • Mfakto
    -v <n> verbosity level: 0=terse, 1=normal, 2=verbose, 3=debug Maximum: 3 (debug)
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    ...ed by Floyd that consists in computing x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub> and so on by iterating the polynomial, and at the same time computing ...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 proper factor of N.
    3 KB (558 words) - 10:28, 6 February 2019
  • Lucas primality test
    Prove that N = 811 is prime knowing that N-1 = 2 &times; 3<sup>4</sup> &times; 5 Let's start with b = 3.
    1 KB (177 words) - 14:31, 17 February 2019
  • Square root
    For example, <math>\sqrt 9 = 3</math> since <math>3^2 = 3 \times 3 = 9</math>. :<math> = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + ...</math>
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Primality testing program
    ...e Prime Freeware] by [[Luke Welsh]] and Yuri Sorkin. Last updated on April 3, 2008.
    2 KB (314 words) - 21:23, 29 August 2019
  • Pépin's test
    ...mber]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1</math> with k = 5 instead of k = 3. ...n>0</math>, <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>.
    2 KB (401 words) - 14:40, 6 March 2019
  • Proof by induction
    *'''Step 3''' *'''Step 3'''
    4 KB (679 words) - 13:57, 20 February 2019
  • M37
    | top5000id=3
    877 bytes (111 words) - 11:04, 18 February 2019
  • GIMPS factoring and sieving
    ...d is eliminated if it was not previously eliminated in the divisibility by 3 test, and so forth. It is not necessary to carry this sieving process all t
    6 KB (962 words) - 10:08, 7 March 2019
  • Modular square root
    ===Modulus congruent to 3 modulo 4=== *Step 2: <math>x = 2</math>, <math>z = 2^7 \bmod 113 = 15</math>, <math>z^{2^3} \bmod 113 = 1</math>, so we have to repeat step 2.
    5 KB (726 words) - 10:38, 6 February 2019
  • Legendre symbol
    ...egin{cases} 1 & \text{if } p \equiv 1 \pmod{4} \\ -1 & \text{if } p \equiv 3 \pmod{4} \end{cases}</math> ...\text{if } p \equiv 1 \text { or } 7 \pmod{8} \\ -1 & \text{if } p \equiv 3 \text{ or } 5 \pmod{8} \end{cases}</math>
    2 KB (348 words) - 18:57, 28 September 2023
  • Law of quadratic reciprocity
    *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 o
    1 KB (208 words) - 18:19, 2 October 2022
  • LLR
    ..." for Gaussian-Mersenne norms and Wagstaff numbers (2<sup>{{V|p}}</sup>+1)/3. The latter uses a strong Fermat PRP-test and the [[Vrba-Reix algorithm]].
    2 KB (300 words) - 22:00, 16 December 2023
  • Double Mersenne number
    *MM(2) = <math>2^3-1</math> = 7, known prime since antiquity *MM(3) = <math>2^7-1</math> = 127, known prime since antiquity
    4 KB (655 words) - 14:50, 19 September 2021
  • P-1 factorization method
    :<math>E = 2^{E_2} * 3^{E_3} * 5^{E_5} * ... * B</math> Then E = 2<sup>3</sup> &times; 3<sup>2</sup> &times; 5 &times; 7 &times; 29.
    5 KB (814 words) - 01:35, 12 March 2019
  • Brent-Suyama extension
    Recall that stage 1 computes S=3<sup>E</sup> where E is the product of prime powers less than B1. Then by [[ A naive stage 2 would then compute T=S<sup>q</sup> = 3<sup>E*q</sup> for successive prime q in the range (B1,B2]. Then p | T-1 if
    2 KB (421 words) - 11:51, 28 January 2019
  • Special number field sieve
    ...eft(\exp\left( \left(\frac{32}{9}n\right)^{\frac{1}{3}} (\log n)^{\frac{2}{3}} \right)\right).</math>
    1 KB (186 words) - 12:07, 19 February 2019
  • Miller-Rabin pseudoprimality test
    *r = 3: <math>463^2\equiv 67\,\pmod{561}</math>
    3 KB (432 words) - 15:33, 28 January 2019
  • Generalized Fermat number
    ...-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. ...zed Fermat numbers the program PFGW can be used. For example to test {{NPr|3|41}} call
    5 KB (726 words) - 09:57, 12 September 2021
  • PrimeGrid
    ...Network Computing]] (BOINC) platform. As of October 2020, there are about 3,300 active participants (on about 16,000 host computers) from 89 countries, ...e Search|321 Prime Search]] searching for mega primes of the form {{Kbn|±|3|2|n}}.
    3 KB (458 words) - 10:28, 26 March 2024
  • 321 Search
    ...oject|distributed computing project]] to search for [[prime]]s of the form 3*2<sup>n</sup>-1.
    1 KB (185 words) - 09:34, 3 August 2021
  • Sieve of Eratosthenes
    ...number which hasn't been crossed out already; these are divisible by <math>3</math> and so are composite.
    4 KB (654 words) - 11:10, 6 February 2019
  • Top GIMPS teams
    | 3 || [[GPUto72|GPU to 72]] || 17738197
    2 KB (206 words) - 09:56, 7 March 2019
  • Aliquot sequence
    ...<math>10^6</math>, 18361 under <math>2*10^6</math>, and 27659 under <math>3*10^6</math>. :<math>2^ \ \ *\ 3</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
    ...that is only divisible by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. ...tive 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
    2 KB (447 words) - 00:22, 10 July 2023
  • Twin prime
    | align="right" | 10<sup>3</sup> || align="right" | 35
    2 KB (255 words) - 06:08, 21 February 2023
  • Computing power
    ...TFLOPS) or petaFLOPS (PFLOPS)) are normally used. [[GIMPS]] has also used 3 other measurements over its lifespan (P90 years, PII-400 years, and GHz-day
    2 KB (257 words) - 22:54, 3 February 2019
  • Math rendering
    | <code>\prime, f^\prime, f', f'', f^{(3)}, \dot y, \ddot y</code> | <math>\prime, f^\prime, f', f'', f^{(3)} \!, \dot y, \ddot y</math>
    33 KB (4,920 words) - 10:54, 7 March 2019
  • Why participate in GIMPS?
    ...like 6 and 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
    7 KB (1,252 words) - 09:47, 7 March 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
  • NewPGen
    *k&times;2<sup>n</sup>+1, k&times;2<sup>n+1</sup>+3 (Sophie Germain) ...>+1 (to test k&times;b<sup>n</sup>+1, and if that is prime check the other 3 for the chance of a twin prime or CC 2nd kind)
    3 KB (529 words) - 09:32, 7 March 2019
  • Sophie Germain prime
    | 2 || 3 || {{NRi|2}} | 3 || 5 || {{NRi|3|1}}
    1 KB (171 words) - 04:26, 3 November 2020
  • Torture test
    ...using 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.
    2 KB (323 words) - 10:59, 18 June 2019
  • Floor function
    | 2 || 3 | 2 || 3
    813 bytes (111 words) - 16:56, 29 August 2022
  • Power of two
    ...so 1024 bytes = 1 kibibyte. 1024<sup>2</sup> bytes = 1 mebibyte, 1024<sup>3</sup> bytes = 1 gibibyte, and so on.
    839 bytes (127 words) - 11:38, 6 February 2019
  • Introductory stuff
    ...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]].
    14 KB (2,370 words) - 15:15, 17 August 2019
  • Glossary
    *'''[[Exponent]]''' - In <math>3^5</math>, 5 is the exponent.
    1 KB (190 words) - 10:55, 7 March 2019
  • Overclocking
    ...your AGP 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
    ...efined 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
  • Pocklington's theorem
    ...est|Selfridge's]] test can be used to prove primality. If <math>f^{10} > n^3</math>, the [[KonyaginPomerance|Konyagin-Pomerance Test]] can be used. If <
    2 KB (346 words) - 19:51, 30 August 2019
  • Keyboard shortcuts
    !rowspan="3"|{{Key|d}}
    3 KB (431 words) - 10:20, 7 March 2019
  • GIMPS type of work
    ...ic curve method]], and work units of this type are typically distributed 1-3 curves at a time from PrimeNet (one can change the number of curves by modi
    4 KB (757 words) - 15:17, 25 July 2020
  • How to get started with GIMPS
    *'''Windows 3.1''': Download ''prime.zip''. This version is not fully automatic, but you ==Step 3==
    4 KB (623 words) - 13:39, 26 March 2019
  • P+1 factorization method
    E = 2<sup>3</sup> &times; 3<sup>2</sup> &times; 5 &times; 7 *Multiply index by 3: <math>427936^2 - 2\equiv 299066</math>, so the first pair is <math>(V_n ,
    8 KB (1,536 words) - 11:35, 12 February 2019

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