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  • 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
  • Legendre symbol
    *1 if <math>a</math> is a square [[modular arithmetic|modulo]] <math>p</math> *&minus;1 if <math>a</math> is not a square modulo <math>p</math>, or in other words
    2 KB (348 words) - 18:57, 28 September 2023
  • Law of quadratic reciprocity
    *If at least one of <math>p</math> or <math>q</math> are congruent to 1 mod 4: <math>p</math> is a quadratic residue modulo <math>q</math> if and o This does not cover the cases where we want to know whether -1 or 2 are quadratic residues or non-residues modulo <math>p</math>.
    1 KB (208 words) - 18:19, 2 October 2022
  • Ten million digits
    ...overy of [[M38]] (the first [[megaprime]] or [[prime]] number greater than 1 million [[decimal]] [[digit]]s) in June of 1999, the next [[EFF prizes]] fo
    979 bytes (146 words) - 14:23, 6 March 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
  • Double Mersenne number
    ...'MM<sub>p</sub>''', '''MMp''', or '''MM(p)''' and refer to <math>2^{2^p-1}-1</math>. Early on it was thought that if M(p) was prime so too was MM(p). *MM(2) = <math>2^3-1</math> = 7, known prime since antiquity
    4 KB (655 words) - 14:50, 19 September 2021
  • CUDAPm1
    ...ram, based on [[CUDALucas]] code, for testing [[P-1 factorization method|P-1]] testing on [[GPU]]s.
    419 bytes (53 words) - 15:51, 28 January 2019
  • P-1 factorization method
    '''P-1''' is a [[:Category:Factorization|factorization method]] invented by John P ...a prime which does not divide the integer ''a'', then <math>a^{p-1}\equiv 1 \mbox{( mod } p)</math>.
    5 KB (814 words) - 01:35, 12 March 2019
  • Brent-Suyama extension
    ...ossibility of finding factors outside the normal range expected from the P-1 bounds. ...than B1. Then by [[Fermat's Little Theorem]], a prime number p | S-1 if p-1 | E.
    2 KB (421 words) - 11:51, 28 January 2019
  • Self-initializing quadratic sieve
    :<math>45^2\,\equiv \,2^4*7^0*13^1</math> :<math>123^2\,\equiv \,2^{10}*7^0*13^1</math>
    10 KB (1,763 words) - 02:56, 12 March 2019
  • Special number field sieve
    :<math>\Theta\left(\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

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