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Create the page "K,n-pair" on this wiki! See also the search results found.

  • Mersenne prime
    ...2}</math>, where <math>S_0=4</math> and for <math>k>0</math>, <math>S_k=S_{k-1}^2-2</math>. * [[Riesel prime 2 1|Riesel primes for k=1]]
    5 KB (857 words) - 14:53, 19 September 2021
  • Fermat number
    *'''[[Proth's theorem]]''' -- (1878) Let {{V|N}} = {{Kbn|+|k|2|m}} with odd {{Vk}} < 2<sup>{{V|m}}</sup>. If there is an integer {{V|a}} ...des <math>S_{2^n-2}</math> where <math>S_0=5</math> and <math>S_{k+1}=S^2_{k}-2</math> ([http://www.robertgerbicz.tar.hu/Fermatnumbers.pdf Proof] by Rob
    12 KB (1,913 words) - 14:35, 9 August 2021
  • Cunningham project
    :<math>(b^{kn}-1) = (b^n-1) \sum _{r=0}^{k-1} b^{rn}</math> for any value of <math>k</math> and
    7 KB (1,150 words) - 23:48, 19 April 2023
  • Lucas-Lehmer test
    ...ath>\omega</math> to get that <math>\omega^{k-j}=1</math>, with <math>1\le k-j\le F^2-1</math>. We have proven that <math>\omega</math> satisfies <math Any number of the form <math>I_k=\omega^k + \bar\omega^k</math>, where <math>k</math> is positive odd integer, can be used as an initial value <math>S_0</
    20 KB (3,572 words) - 14:30, 17 February 2019
  • 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 Assist in proving the [[Liskovets-Gallot conjectures]] for the forms {{Kbn|±|k|2|n}} where {{Vn}} is always odd '''and''' where {{Vn}} is always even.
    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> ...Tukey FFTs, for example), via the identity <math>jk = -(j-k)^2/2 + j^2/2 + k^2/2</math>.
    17 KB (2,684 words) - 18:50, 28 September 2023
  • No Prime Left Behind
    ...n 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 reser
    745 bytes (111 words) - 02:17, 1 May 2024
  • Rational number
    :<math>\large a + \frac{k(b-a)}{n+1}</math> by varying the number <math>k</math> from 1 to <math>n</math>. Then we can make the value <math>n</math>
    3 KB (541 words) - 15:01, 26 March 2023
  • Modular arithmetic
    ...>2^m</math>, let the number to be inverted be <math>N</math> and let <math>k = \log_2 m</math> rounded to the next integer. Then the method is: # Perform k times: Set x = x(2-Nx) mod 2^m
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    ...] \,+\, a'[1] * 2^k \,+\, a'[2] * 2^{2k} \,+\, ... \,+\, a'[s-1] * 2^{(s-1)k}</math>. and similarly for b' and c'. The number <math>k</math> is the number of bits of the integer, and <math>n = ks</math>.
    4 KB (582 words) - 17:01, 29 August 2022
  • Elliptic curve method
    ...ion does not change by multiplying all coordinates by the same value <math>k\neq 0</math>. ...y using another point. When we compute k'''P''' we will be also computing (k+1)'''P'''.
    19 KB (3,181 words) - 22:27, 6 July 2023
  • Wacław Sierpiński
    ...ntegers {{Vk}} (named [[Sierpiński number]]s after him) such that {{Kbn|+|k|n}} is composite for all {{Vn}}.
    592 bytes (86 words) - 00:38, 15 January 2024
  • Sierpiński number
    ...ierpiński number''' is an odd natural number {{Vk}} such that all {{Kbn|+|k|n}} for all {{Vn}} are [[Composite number|composite]].
    324 bytes (48 words) - 13:37, 8 April 2023
  • Sierpiński problem
    Consider numbers of the form {{V|N}} = {{Kbn|+|k|n}}, where {{Vk}} is odd and {{Vn}} > 0. If, for some fixed {{Vk}}, every i
    5 KB (650 words) - 10:25, 26 March 2024
  • Seventeen or Bust
    |result=11 k's eliminated as a standalone project, 1 k eliminated as a subproject on PrimeGrid ...he 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 [[Sierpiński number]]s
    3 KB (544 words) - 16:44, 21 July 2019
  • Perfect number
    :where ''q'', ''p''<sub>1</sub>, &hellip;, ''p''<sub>''k''</sub> are distinct primes and ''q'' &equiv; &alpha; &equiv; 1 (mod 4) (Eu
    6 KB (885 words) - 11:33, 7 March 2019
  • Mfaktc
    *[[Factor]] sizes <math><2^{95}</math> and '''[[value k]]''' <math><2^{63.9}</math>
    5 KB (765 words) - 14:54, 25 February 2019
  • Mfakto
    ...ssSize: defines how far many bits of the sieve each TF block processes (in K bits). Larger values may lead to less wasted cycles by reducing the number
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Riesel number
    A '''Riesel number''' is a value of ''k'' such that {{Kbn|k|n}} is always composite for all [[natural number]]s.
    827 bytes (112 words) - 08:21, 25 March 2024
  • 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 [[distrib
    3 KB (501 words) - 05:20, 3 August 2021
  • Primality testing program
    | k &times; b<sup>n</sup>±c general numbers | k &times; b<sup>n</sup>±c
    2 KB (314 words) - 21:23, 29 August 2019
  • Pépin's test
    ...t 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
    :<math>\sum_{k=1}^{n}\,(2k-1)\,=\,n^{2}</math> ::<math>\sum_{k=1}^{1}\,(2k-1)\,=\,2-1\,=\,1^{2}</math>
    4 KB (679 words) - 13:57, 20 February 2019
  • GIMPS factoring and sieving
    ...e of the algebraic form <math>2kp+1</math> for some positive integer <math>k</math> and furthermore must also leave a remainder of either 1 or 7 upon di
    6 KB (962 words) - 10:08, 7 March 2019
  • LLR
    **[[Lucas-Lehmer-Riesel algorithm]] for {{Kbn|k|n}} numbers. **[[Proth's theorem|Proth algorithm]] for {{Kbn|+|k|n}} numbers.
    2 KB (300 words) - 22:00, 16 December 2023
  • P-1 factorization method
    ...1</math>, where ''k'' is a non-negative integer. This means that <math>a^{k(p-1)} - 1</math>
    5 KB (814 words) - 01:35, 12 March 2019
  • Self-initializing quadratic sieve
    ...operatorname{tlog}[p]</math> to all locations <math>\operatorname{soln1} + k*p</math> of the sieve array. ...operatorname{tlog}[p]</math> to all locations <math>\operatorname{soln2} + k*p</math> of the sieve array.
    10 KB (1,763 words) - 02:56, 12 March 2019
  • Miller-Rabin pseudoprimality test
    ...er being tested for primality, and <math>N = 2^n\,k + 1</math> where <math>k</math> is an odd number. Then consider the sequence <math>a^k</math>, <math>a^{2k}</math>, <math>a^{4k}</math>..., <math>a^{N-1}</math> m
    3 KB (432 words) - 15:33, 28 January 2019
  • Generalized Fermat number
    ...F_{n,2}</math> numbers can be proven prime by using [[Pépin's test]] with k=5. ...[[Generalized Fermat number]]s for any [[Proth prime|Proth primes {{Kbn|+|k|n}}]] are listed as ''GF Divisor'' on their own page. They are listed as ''
    5 KB (726 words) - 09:57, 12 September 2021
  • PrimeGrid
    ...rime Search|Proth Prime Search]]: searching for primes of the form {{Kbn|+|k|2|n}}. ...ed|Proth Prime Search Extended]]: searching for primes of the form {{Kbn|+|k|2|n}}.
    3 KB (458 words) - 10:28, 26 March 2024
  • Sieve of Eratosthenes
    *First, list out all the integers <math>1 \leq k \leq N</math>. ...ssary because all composite numbers <math>11 \geq \sqrt{N} = 10 \geq \sqrt{k}</math> have already been crossed out.
    4 KB (654 words) - 11:10, 6 February 2019
  • Twin Prime Search
    ...t that looks for large [[twin prime]]s ([[Riesel prime|Riesel type]] {{Kbn|k|n}}) of world record size.
    826 bytes (100 words) - 11:04, 21 March 2024
  • Riesel Prime Search
    :The project is searching for [[Riesel prime]]s {{Kbn|k|n}}, {{Vk}} > 1.
    845 bytes (120 words) - 02:21, 1 May 2024
  • Seventeen or Bust/Factoring
    factor | k*2^n+1
    3 KB (491 words) - 13:39, 1 February 2019
  • Seventeen or Bust/PRP
    ==Working on the k,n pair==
    888 bytes (145 words) - 13:40, 1 February 2019
  • Sieving
    ...ts of the array <math>k</math>, <math>k+p</math>, <math>k+2p</math>, <math>k+3p</math>, and so on correspond to multiples of the prime.
    3 KB (521 words) - 11:14, 6 February 2019
  • Sieving program
    ...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 [[TPSieve]] (similar to PPSieve, but for *[[TwinGen]] (performing sieving of numbers of the form k&times;2<sup>n</sup>+/-1) http://www.underbakke.com/primes/
    2 KB (220 words) - 11:42, 7 March 2019
  • NewPGen
    *k&times;b<sup>n</sup>+1 *k&times;b<sup>n</sup>-1
    3 KB (529 words) - 09:32, 7 March 2019
  • 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</
    549 bytes (88 words) - 18:15, 28 September 2023
  • Proth prime
    ...f numbers, but primes in the form {{Kbn|+|k|n}} with 2<sup>''n''</sup> > ''k'' are often called Proth primes. *[[Proth prime table|Table]] with all available ''k''-values
    656 bytes (91 words) - 07:02, 31 August 2020
  • Riesel prime
    ...ial definition 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. ...mersenneforum.org/showthread.php?t=29635 "Team drive #1 for {{Vk}}<300: 26 k's for {{Vn}}>2M"]: [https://www.mersenneforum.org/showpost.php?p=655608 #1
    2 KB (279 words) - 03:48, 24 April 2024
  • Pocklington's theorem
    :Let <math>N-1 = q^k\,R</math> where ''q'' is a prime which does not divide ''R''. If there is a
    2 KB (346 words) - 19:51, 30 August 2019
  • Multiple polynomial quadratic sieve
    ...operatorname{tlog}[p]</math> to all locations <math>\operatorname{soln1} + k*p</math> of the sieve array. ...operatorname{tlog}[p]</math> to all locations <math>\operatorname{soln2} + k*p</math> of the sieve array.
    6 KB (1,068 words) - 14:33, 13 February 2019
  • Proth number
    :{{V|N}} = {{Kbn|+|k|2|n}}
    670 bytes (104 words) - 10:59, 9 July 2021
  • Worktodo.txt
    *[[PRP]]=k,b,n,c<nowiki>[</nowiki>,how_far_factored,tests_saved<nowiki>][</nowiki>,kno *Pfactor=assignment ID,[[value k|k]],b ([[base]]),n ([[exponent]]),c,how far factored,[[Lucas-Lehmer test|LL]]
    2 KB (273 words) - 22:58, 11 May 2019
  • P+1 factorization method
    ...e. When we compute <math>V_{kn}</math> we will be also computing <math>V_{(k+1)n}</math>. ...d when the bit equals one we compute (<math>V_{(2k+1)n}</math>, <math>V_{2(k+1)n}</math>). In both cases we need one duplication and one addition requir
    8 KB (1,536 words) - 11:35, 12 February 2019
  • SNFS polynomial selection
    ...-1)} + a\cdot a^{k-1} + 1</math> and can convert to a quartic in <math>a^{(k-1)/2}</math>. If you want a sextic, tweak the exponents accordingly by 2., ...+ a^{3k} + a^{2k} + a^k + 1</math>, which is a useable quartic in <math>a^k</math>.
    7 KB (1,238 words) - 16:14, 12 February 2019
  • Gerbicz error checking
    Let <math>p</math> be a Proth number, and let <math>k</math>, <math>n</math>, and <math>a</math> be defined as on [[Proth's theor # <math>u(t) \equiv (a^k)^{2^t} \pmod{p}</math>
    3 KB (528 words) - 14:59, 3 October 2023
  • 15k Search
    :This was the initial phase, to gather information about heavy k, for later use with fixed n forms. :Fixed n searches are faster than fixed k, and can be used in conjunction with heavy 15k.
    3 KB (517 words) - 14:51, 15 February 2019
  • Sierpiński number base 5
    A '''Sierpiński number base 5''' is a value of {{Vk}} such that {{Kbn|+|k|5|n}} is always [[composite number|composite]].
    556 bytes (83 words) - 10:57, 14 October 2020

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