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  • Residue
    ...of the L-L test. When the numbers being tested are large: <math>\gt2^{64}-1</math> (i.e. exponents larger than 64) and above, the residue is 16 hexadec Here is the Lucas test for <math>2^7-1</math>, which is 127:
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    ! LL test !! PRP test !! Trial factoring !! ECM factoring !! P-1 factoring ...mount of factoring work using (say) Pollard's [[p-1 factorization method|p-1 method]] or the [[elliptic curve method]], both of which involve manipulati
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...CPU resources. However GPU sieving is not supported on compute capability 1.x GPUs in 0.20. Mfaktc 0.21 can do GPU sieving for those old GPUs. *A [[CUDA]] capable GPU with compute capability 1.1 (or newer), any Geforce 8000, 9000, 200, 400, 500 series ''except'' those w
    5 KB (765 words) - 14:54, 25 February 2019
  • Mfakto
    -v <n> verbosity level: 0=terse, 1=normal, 2=verbose, 3=debug -st run built-in selftest (about 1,500 testcases) and exit
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    *x<sub>1</sub> = f(x<sub>0</sub>) *x<sub>2</sub> = f(x<sub>1</sub>)
    3 KB (558 words) - 10:28, 6 February 2019
  • Riesel number
    ...ether 509203 is the smallest Riesel number or not (the '''[[Riesel problem 1]]'''), a [[distributed computing project]] was created named [[Riesel Sieve *[[Riesel problem 1]]
    827 bytes (112 words) - 08:21, 25 March 2024
  • M12
    ...t [[University of California, Los Angeles]] found [[M13]], 2<sup>521</sup>-1. .../sup>-1=7) also produces a prime. When this value is tested (2<sup>7</sup>-1=127), another prime is produced. So, Lucas was testing to see if this trend
    2 KB (354 words) - 14:52, 19 September 2021
  • Primality test
    *[[Lucas primality test|Lucas Test]]: Used when the number {{V|N}}-1 is completely factored. ...factors of the input number - 1 are known (the unfactored part of {{V|N}}-1 must be less than the [[square root]] of {{V|N}}).
    3 KB (501 words) - 05:20, 3 August 2021
  • Lucas primality test
    ...whether a number N is prime or not, using the complete factorization of N-1. ...>(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
  • Square root
    ...gth of the [[diagonal]] of a [[square (geometry)|square]] with side length 1. ...s to the equation <math>x = \sqrt{x}</math> The solution set is <math>\{0, 1\}</math>.
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Pépin's test
    ...s, 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^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 1''' ...of n, say n<sub>0</sub>. This value n<sub>0</sub> is normally taken to be 1, but that is not essential. In some proofs (see example 4 below) we have to
    4 KB (679 words) - 13:57, 20 February 2019
  • Probable prime
    .... If the result is different from 1, <math>n</math> is composite. If it is 1, <math>n</math> may or may not be prime; <math>n</math> is then called a (w
    2 KB (232 words) - 07:28, 12 March 2024
  • Pseudoprime
    ...ns 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
  • Complex number
    ...> we get <math>(-1, 0)</math>. Since no real number is the square root of -1, we can now understand why the second element is the imaginary part. ...<math>z = x + iy</math>. From the previous paragraph we get: <math>i^2 = -1</math>.
    2 KB (280 words) - 14:59, 26 March 2023
  • Operation Billion Digits
    ...al project; [[p-1 factorization method|p-1]], [[p+1 factorization method|p+1]], and [[Elliptic curve method|ECM]] tests are also unfeasible because they ...nd output its progress by k, d and bit depth. The k is, well, the k on 2kp+1, where p is the exponent you're searching. The d is the divisor tried and t
    6 KB (918 words) - 16:28, 24 July 2020
  • M37
    ...he 37th [[Mersenne prime]]. Specifically it is 2<sup>{{Num|3021377}}</sup>-1. This number was discovered to be [[prime]] on 1988-01-27 by [[Roland Clark
    877 bytes (111 words) - 11:04, 18 February 2019
  • Team Prime Rib
    ...0 CPU years of [[GIMPS factoring and sieving#Trial_factoring|factoring]] (#1 on [[PrimeNet]]).
    620 bytes (88 words) - 11:46, 12 February 2019
  • GIMPS factoring and sieving
    ...re capable of doing three different types of factoring: trial factoring, p-1 and ECM, but only the first two of these types are routinely done on GIMPS ...teger <math>k</math> and furthermore must also leave a remainder of either 1 or 7 upon division by 8. See below for an explanation.
    6 KB (962 words) - 10:08, 7 March 2019
  • Modular square root
    ...eger]] number <math>a</math> modulo an integer <math>m</math> greater than 1 is an integer such that: ...s odd, we assume that the quantity <math>a^{(m-1)/2} \bmod m</math> equals 1 (otherwise there is no square root if <math>a \not\equiv 0\ \pmod m</math>)
    5 KB (726 words) - 10:38, 6 February 2019

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