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• Williams prime
  ...er''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1. Lists of primes for bases ''b'' and ''n''-values can be found here:
  5 KB (744 words) - 07:30, 5 August 2019
• 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
  :<math>(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+ ... +2^{(b-1)a}\right)=2^{ab}-1</math>
  5 KB (857 words) - 14:53, 19 September 2021
• Fermat number
  ...} &equiv; (2^{{V|a}})^{{V|b}} + 1 &equiv; (-1)^{{V|b}} + 1 &equiv; 0 (mod 2^{{V|a}} + 1).) In other words, every :<math>F_{n} = (F_{n-1}-1)^{2}+1</math>
  12 KB (1,913 words) - 14:35, 9 August 2021
• Home prime
  Notations for other bases of 10 are HP_base(n) for base ''b''.
  980 bytes (143 words) - 13:22, 6 March 2019
• Cunningham project
  ...e [[factorization]] of numbers of the form <math>b^n\pm 1</math> for <math>b</math> = 2, 3, 5, 6, 7, 10, 11, 12. The values of the exponent <math>n</mat ...split into two parts directly, for certain combination of values of <math>b</math> and <math>n</math>.
  7 KB (1,150 words) - 23:48, 19 April 2023
• Lucas-Lehmer test
  ...ber denoted S_n where S₀=4 and S_n = (S_n-1)² &minus; 2. In 1930, the American mathematician [[Derrick ...th> is the element) must divide N-1. Now exhibit an element whose order is N-1 and we are done.
  20 KB (3,572 words) - 14:30, 17 February 2019
• Multiplication
  ...elf b times. The result of a multiplication is called the product of a and b, and each of the numbers is called a [[factor]] of the product ab. Multiplication is denoted a&times;b,(a)(b), or simply ab. The symbol '&times;' is known as the multiplication sign.
  2 KB (271 words) - 17:00, 29 August 2022
• Conjectures 'R Us
  For every base ({{Vb}} &le; 1030) for the forms {{Kbn|±|k|b|n}} there is a {{Vk}}-value for each form that has been conjectured to be t
  3 KB (503 words) - 02:20, 1 May 2024
• Rational number
  ...h> where <math>a</math> (the '''numerator''') is any [[integer]] and <math>b</math> (the '''denominator''') is an integer different from zero. The set o The notation <math>a/b</math> is called '''fraction'''. A fraction is irreducible when both number
  3 KB (541 words) - 15:01, 26 March 2023
• Repunit
  :<math>(b^n-1)\over (b-1)</math>.
  1 KB (207 words) - 08:04, 12 March 2024
• Modular arithmetic
  ...ers representing the hours 0, 1, 2, 3,..., 11 is known as <b>Z</b>/12<b>Z</b>. ...o numbers ''A'' and ''B'' are said to be congruent modulo ''n'' if ''A''-''B'' is a multiple of ''n''.
  4 KB (625 words) - 10:25, 23 January 2019
• Montgomery multiplication
  The Montgomery multiplication computes <math>c'\,=\,2^{-n}\,a'b'\bmod{m}</math>. Notice that <math>pq\,\equiv \, r</math> implies <math>p'q ...e perform the actual Montgomery multiplication <math>c' \,\equiv \,2^{-n}a'b'</math>.
  4 KB (582 words) - 17:01, 29 August 2022
• Elliptic curve method
  ...expression to <math>y^2\,\equiv \,x^3\,+\,ax\,+\,b</math> (where ''a'', ''b'', ''x'' and ''y'' are integers like 3 or -12, and the sign <math>\equiv</m which is an elliptic curve when <math>B\neq 0</math> and <math>A\neq \pm 2</math>.
  19 KB (3,181 words) - 22:27, 6 July 2023
• Mfakto
  %d - date (Mon nn) "%b %d" (strftime format)
  17 KB (2,524 words) - 12:39, 24 January 2019
• Lucas primality test
  ...es whether a number N is prime or not, using the complete factorization of N-1. ...up>(N-1)/q</sup> is not congruent to 1 modulo N for any prime divisor q of N-1, then N is a prime.
  1 KB (177 words) - 14:31, 17 February 2019
• Primality testing program
  | [[Mersenne number]]s<br/>a &times; bⁿ±c (only factoring and [[probable prime|PRP]]-testing) | k &times; bⁿ±c general numbers
  2 KB (314 words) - 21:23, 29 August 2019
• LLR
  **{{V|N}}-1 [[Pocklington algorithm]] for {{Kbn|+|k|b|n}} numbers. **{{V|N}}+1 [[Morrison algorithm]] for {{Kbn|k|b|n}} numbers.
  2 KB (300 words) - 22:00, 16 December 2023
• P-1 factorization method
  :<math>E = 2^{E_2} * 3^{E_3} * 5^{E_5} * ... * B</math> ...ath>2^{E_2}</math> is about B1 and the same for the other prime numbers. ''B'' is the greatest prime number less than or equal to B1.
  5 KB (814 words) - 01:35, 12 March 2019
• Self-initializing quadratic sieve
  ...t^2\equiv u\,\pmod N</math> where <math>t = ax+b</math> and <math>u = g_{a,b}(x)</math>. ...agraph does not depend on the particular values of <math>a</math> or <math>b</math>, so the factor base can be computed at the beginning of SIQS. The up
  10 KB (1,763 words) - 02:56, 12 March 2019
• Generalized Fermat number
  ...e built a list of large primes of the form: b^{2^m}+1, ''b &ge; 2'' and ''m &ge; 1''. ...primes of the form: a^{2^m}+b^{2^m}, ''b > a &ge; 2'' and ''m &ge; 1''.
  5 KB (726 words) - 09:57, 12 September 2021
• PrimeGrid
  ...eralized Fermat Prime Search]]: searching for primes of the form {{Kbn|+|1|b|2^{{Vn}}}}. ...eralized Cullen Prime Search]]: searching for primes of the form {{Kbn|+|n|b|n}}.
  3 KB (458 words) - 10:28, 26 March 2024
• Aliquot sequence
  :<math>s_n = \sigma(s_{n-1}) - s_{n-1}</math> :<math>N=p^{a}*q^{b}*r^{c}</math>. . .
  6 KB (914 words) - 19:49, 21 February 2023
• Math rendering
  | <code>\exp_a b = a^b, \exp b = e^b, 10^m</code> | <math>\exp_a b = a^b, \exp b = e^b, 10^m</math>
  33 KB (4,920 words) - 10:54, 7 March 2019
• Sieving
  ...nce forms an arithmetic progression, so it has the form <math>m\, =\, an + b</math>, while in the second the sequence has the form <math>m\, =\, an^2 + ...n</math> is the element number of the array. We have to compute <math>k = -b/a \,\pmod p</math>. Then the elements of the array <math>k</math>, <math>k+
  3 KB (521 words) - 11:14, 6 February 2019
• Sieving program
  ...ieve]] (performing sieving of generalized Cullen/Woodall numbers n &times; bⁿ+-1) http://sites.google.com/site/geoffreywalterreynolds/program *[[AthGFNSieve]] (performing sieving of generalized Fermat numbers b^2ⁿ+1) http://www.underbakke.com/AthGFNsv/
  2 KB (220 words) - 11:42, 7 March 2019
• NewPGen
  *k&times;bⁿ+1 *k&times;bⁿ-1
  3 KB (529 words) - 09:32, 7 March 2019
• Multiple polynomial quadratic sieve
  ...t^2\equiv u\,\pmod N</math> where <math>t = ax+b</math> and <math>u = g_{a,b}(x)</math>. ...agraph does not depend on the particular values of <math>a</math> or <math>b</math>, so the factor base can be computed at the beginning of MPQS. The up
  6 KB (1,068 words) - 14:33, 13 February 2019
• GIMPS chances of finding a prime number
  ...ed to the default limits for exponent p, that it has 1 chance in ( p / ( (b-1) x 1.803 )) of being prime.
  3 KB (593 words) - 10:09, 7 March 2019
• Worktodo.txt
  *[[PRP]]=k,b,n,c<nowiki>[</nowiki>,how_far_factored,tests_saved<nowiki>][</nowiki>,known *Pfactor=assignment ID,[[value k|k]],b ([[base]]),n ([[exponent]]),c,how far factored,[[Lucas-Lehmer test|LL]] tes
  2 KB (273 words) - 22:58, 11 May 2019
• P+1 factorization method
  <math>\large U_n = uU_{n-1} - U_{n-2}\,,\, V_n = uV_{n-1} - V_{n-2}</math> for <math>n\geq 2</math> ...iplication explained above can be translated to pseudocode as follows (let B = V_n):
  8 KB (1,536 words) - 11:35, 12 February 2019
• SNFS polynomial selection
  ...resulting polynomial is symmetric: (for the sake of less typing, let <math>b=a^k</math>) ...b^{11}-1)/(b-1) = b^{10} + b^9 + b^8 + b^7 + b^6 + b^5 + b^4 + b^3 + b^2 + b + 1</math>
  7 KB (1,238 words) - 16:14, 12 February 2019
• Aurifeuillian factor
  ...rs of the form <math>b^n - 1</math> or <math>\Phi_n(b)</math>, where <math>b = s^2 \cdot t</math> with [[Square-free integer|square-free]] <math>t</math ...d 4, <math>b^n-1</math> have aurifeuillean factorization, otherwise, <math>b^n+1</math> have aurifeuillean factorization.
  10 KB (1,257 words) - 08:04, 24 June 2019
• Cullen number
  ...generalized Cullen number''' base {{Vb}} is a number of the form {{Kbn|+|n|b|n}}. '''(perhaps own page?)'''
  2 KB (252 words) - 17:39, 31 August 2021
• Factoring program
  ...d MacOSX) (performing ECM, P-1, and P+1 to search for factors of a &times; bⁿ ± c) ftp://mersenne.org/gimps/, also http://mersenneforum.org/
  2 KB (273 words) - 21:53, 8 July 2023
• Mtsieve
  ...factors of numbers in the form {{Kbn|+|k|b|n}} and {{Kbn|k|b|n}} for fixed b and n and variable k *fkbnsieve: search for factors of the form k*bⁿ+c for fixed k, b, and n and variable c
  2 KB (338 words) - 06:58, 28 March 2023
• Carol-Kynea prime
  ...nea numbers''' are numbers of the form <math>(b^n-1)^2-2</math> and <math>(b^n+1)^2-2</math>, respectively, while '''Carol primes''' and '''Kynea primes ...<math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math>. A Carol/Kynea prime is a [[prime]] which has one of the ab
  8 KB (1,172 words) - 00:38, 6 July 2023
• Williams prime MM least
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b-1)|b|n}}, least {{Vn}}-values}} ...Vb}} ≤ 2049 which generates a [[Williams prime]] of the form {{Kbn|(b-1)|b|n}}.
  2 KB (211 words) - 07:40, 1 August 2021
• Sieving a range of sequences
  Sieving a range {{Kbn|3|2|n}} to {{Kbn|20|19|n}} (in general {{Kbn|(b+1)|b|n}}) create a batch file named "run.bat" with: srsieve -G -n 1 -N 10000 -P 3000000000 "%b1%*%base%^n-1"
  775 bytes (124 words) - 08:39, 22 April 2019
• LLR testing a range of sequences
  Testing a range {{Kbn|3|2|n}} to {{Kbn|20|19|n}} (in general {{Kbn|(b+1)|b|n}}) create a batch file named "run.bat" with:
  871 bytes (147 words) - 12:29, 23 April 2019
• Woodall prime
  ...a Woodall prime with a general base {{Vb}} &gt; 2, so of the form {{Kbn|n|b|n}}. ...ime M|{{Kbn|(n-1)|b|n}}]]''' or '''[[:Category:Woodall prime P|{{Kbn|(n+1)|b|n}}]]'''.
  1 KB (205 words) - 09:53, 13 March 2024
• Cullen prime
  ...a Cullen prime with a general base {{Vb}} &gt; 2, so of the form {{Kbn|+|n|b|n}}. For a complete list in this Wiki the primes of the forms {{Kbn|+|n|b|n}} are given with {{Vb}} &ge; 2, {{Vn}} &ge; 1 and listed as "Cullen prime
  2 KB (286 words) - 17:22, 7 May 2023
• Williams prime MM table
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b-1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n
  460 bytes (65 words) - 23:05, 31 July 2021
• Williams prime MP table
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n
  462 bytes (65 words) - 08:05, 1 August 2021
• Williams prime PM table
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b+1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n
  458 bytes (65 words) - 14:02, 1 August 2021
• Williams prime PP table
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}}} |listseparators={¦class="wikitable"\n!b\n!style="width:9%"¦Last&nbsp;update\n!max&nbsp;n\n!#\n!Primes for n,,,\n¦
  447 bytes (65 words) - 08:12, 18 June 2019
• Williams prime MP least
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}, least {{Vn}}-values}} ...}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b-1)|b|n}}.
  4 KB (457 words) - 11:14, 11 May 2024
• Williams prime PM least
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b+1)|b|n}}, least {{Vn}}-values}} ...Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|(b+1)|b|n}}.
  1 KB (161 words) - 13:57, 1 August 2021
• Williams prime PP least
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}, least {{Vn}}-values}} ...}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b+1)|b|n}}.
  2 KB (306 words) - 09:54, 7 September 2020
• Williams prime MM remaining
  {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b-1)|b|n}}, remaining data}} ...yet available as separate page for Williams primes of the form {{Kbn|(b-1)|b|n}} for {{Vb}} < 2050.
  962 bytes (137 words) - 07:42, 1 August 2021

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