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Create the page "(b-1)*b^n-1" on this wiki! See also the search results found.
- ...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
- <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
- :<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
- ...} ≡ (2<sup>{{V|a}}</sup>)<sup>{{V|b}}</sup> + 1 ≡ (-1)<sup>{{V|b}}</sup> + 1 ≡ 0 (mod 2<sup>{{V|a}}</sup> + 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
- Notations for other bases of 10 are HP<sub>base</sub>(n) for base ''b''.980 bytes (143 words) - 13:22, 6 March 2019
- ...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
- ...ber denoted S<sub>n</sub> where S<sub>0</sub>=4 and S<sub>n</sub> = (S<sub>n-1</sub>)<sup>2</sup> − 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
- ...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×b,(a)(b), or simply ab. The symbol '×' is known as the multiplication sign.2 KB (271 words) - 17:00, 29 August 2022
- For every base ({{Vb}} ≤ 1030) for the forms {{Kbn|±|k|b|n}} there is a {{Vk}}-value for each form that has been conjectured to be t3 KB (503 words) - 02:20, 1 May 2024
- ...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 number3 KB (541 words) - 15:01, 26 March 2023
- :<math>(b^n-1)\over (b-1)</math>.1 KB (207 words) - 08:04, 12 March 2024
- ...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
- 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
- ...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
- %d - date (Mon nn) "%b %d" (strftime format)17 KB (2,524 words) - 12:39, 24 January 2019
- ...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
- | [[Mersenne number]]s<br/>a × b<sup>n</sup>±c (only factoring and [[probable prime|PRP]]-testing) | k × b<sup>n</sup>±c general numbers2 KB (314 words) - 21:23, 29 August 2019
- **{{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
- :<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
- ...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 up10 KB (1,763 words) - 02:56, 12 March 2019
- ...e built a list of large primes of the form: b<sup>2<sup>m</sup></sup>+1, ''b ≥ 2'' and ''m ≥ 1''. ...primes of the form: a<sup>2<sup>m</sup></sup>+b<sup>2<sup>m</sup></sup>, ''b > a ≥ 2'' and ''m ≥ 1''.5 KB (726 words) - 09:57, 12 September 2021
- ...eralized Fermat Prime Search]]: searching for primes of the form {{Kbn|+|1|b|2<sup>{{Vn}}</sup>}}. ...eralized Cullen Prime Search]]: searching for primes of the form {{Kbn|+|n|b|n}}.3 KB (458 words) - 10:28, 26 March 2024
- :<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
- | <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
- ...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
- ...ieve]] (performing sieving of generalized Cullen/Woodall numbers n × b<sup>n</sup>+-1) http://sites.google.com/site/geoffreywalterreynolds/program *[[AthGFNSieve]] (performing sieving of generalized Fermat numbers b<sup>2<sup>n</sup></sup>+1) http://www.underbakke.com/AthGFNsv/2 KB (220 words) - 11:42, 7 March 2019
- *k×b<sup>n</sup>+1 *k×b<sup>n</sup>-13 KB (529 words) - 09:32, 7 March 2019
- ...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 up6 KB (1,068 words) - 14:33, 13 February 2019
- ...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
- *[[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]] tes2 KB (273 words) - 22:58, 11 May 2019
- <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<sub>n</sub>):8 KB (1,536 words) - 11:35, 12 February 2019
- ...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
- ...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
- ...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
- ...d MacOSX) (performing ECM, P-1, and P+1 to search for factors of a × b<sup>n</sup> ± c) ftp://mersenne.org/gimps/, also http://mersenneforum.org/2 KB (273 words) - 21:53, 8 July 2023
- ...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*b<sup>n</sup>+c for fixed k, b, and n and variable c2 KB (338 words) - 06:58, 28 March 2023
- ...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 ab8 KB (1,172 words) - 00:38, 6 July 2023
- {{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 {{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
- 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
- ...a Woodall prime with a general base {{Vb}} > 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
- ...a Cullen prime with a general base {{Vb}} > 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}} ≥ 2, {{Vn}} ≥ 1 and listed as "Cullen prime2 KB (286 words) - 17:22, 7 May 2023
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b-1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n460 bytes (65 words) - 23:05, 31 July 2021
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n462 bytes (65 words) - 08:05, 1 August 2021
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|(b+1)|b|n}}}} |resultsheader=<b>There are %PAGES% sequences</b>:\n458 bytes (65 words) - 14:02, 1 August 2021
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}}} |listseparators={¦class="wikitable"\n!b\n!style="width:9%"¦Last update\n!max n\n!#\n!Primes for n,,,\n¦447 bytes (65 words) - 08:12, 18 June 2019
- {{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
- {{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
- {{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
- {{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
- {{DISPLAYTITLE:Riesel primes of the form {{Kbn|k|b|n}}, least {{Vn}}-values}} ...the least {{Vn}} ≥ 1 generating a [[Riesel prime]] of the form {{Kbn|k|b|n}} for 2 ≤ {{Vb}} ≤ 1030 and 2 ≤ {{Vk}} ≤ 12.6 KB (684 words) - 09:40, 17 March 2024
- {{DISPLAYTITLE:Proth primes of the form {{Kbn|+|k|b|n}}, least ''n''-values}} ...≥ 1 generating a [[Proth prime]] of the form {{Kbn|+|k|b|n}} for 2 ≤ ''b'' ≤ 1030 and 2 ≤ ''k'' ≤ 12.7 KB (795 words) - 08:03, 5 May 2024
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}, remaining data}} ...as separate page for Williams primes of the form {{Kbn|+|(b+1)|b|n}} for ''b'' < 1025.962 bytes (135 words) - 07:49, 26 June 2019
- {{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}} < 1025.962 bytes (136 words) - 14:00, 1 August 2021
- {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}, remaining data}} ...t available as separate page for Williams primes of the form {{Kbn|+|(b-1)|b|n}} for {{Vb}} < 1025.966 bytes (138 words) - 08:03, 1 August 2021
- * {{Kbn|±|k|b|n}} (by default {{Vb}} is 2 but it can be changed) * {{Kbn|±|n|b|n}} ([[Cullen prime|Generalized Cullen]]/[[Woodall prime|Generalized Woodal667 bytes (101 words) - 16:44, 31 August 2021
- It can be used for every sequence {{Kbn|k|b|n}} and {{Kbn|+|k|b|n}} without limitations of the {{Vk}}-value. nash - a tool for computing Nash weights for sequences k*b^n+-12 KB (330 words) - 09:11, 23 September 2021
- | [[:Category:Riesel prime|Riesel primes {{Kbn|k|b|n}}]], {{Vb}}>2 || base || style="text-align:right;"|{{Num|{{#expr:{{PAG | [[:Category:Proth prime|Proth primes {{Kbn|+|k|b|n}}]], {{Vb}}>2 || base || style="text-align:right;"|{{Num|{{#expr:{{PAG11 KB (1,385 words) - 17:23, 5 April 2024
- ...s to find [[Cullen prime|Generalized Cullen primes]] of the form {{Kbn|+|n|b|n}} for bases {{Vb}} which have no known primes. |resultsheader=<b>The bases currently being searched by the project</b>:2 KB (203 words) - 17:29, 7 May 2023
- ===Create a new page for a Riesel-type entry {{Kbn|k|b|n}}=== ===Create a new page for a Proth-type entry {{Kbn|+|k|b|n}}===2 KB (321 words) - 11:03, 23 April 2024
- :<math>(b^n-1)\over (b-1)</math>. |resultsheader=<b>There are %PAGES% sequences</b>:\n1 KB (177 words) - 09:22, 6 April 2024