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  2. Fermat primality test - Wikipedia

    en.wikipedia.org/wiki/Fermat_primality_test

    The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Concept ... Either 221 is prime, or 38 is a Fermat liar, ...

  3. Fermat number - Wikipedia

    en.wikipedia.org/wiki/Fermat_number

    If 2 k + 1 is prime and k > 0, then k itself must be a power of 2, [1] so 2 k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023 [update] , the only known Fermat primes are F 0 = 3 , F 1 = 5 , F 2 = 17 , F 3 = 257 , and F 4 = 65537 (sequence A019434 in the OEIS ).

  4. Primality test - Wikipedia

    en.wikipedia.org/wiki/Primality_test

    The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows: Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.

  5. Fermat's little theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_little_theorem

    Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem. [3]

  6. Pépin's test - Wikipedia

    en.wikipedia.org/wiki/Pépin's_test

    In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test . The test is named after a French mathematician, Théophile Pépin .

  7. Probable prime - Wikipedia

    en.wikipedia.org/wiki/Probable_prime

    An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a (p−1)/2 equals () modulo p, where () is the Jacobi symbol. An Euler probable prime which is composite is called an Euler–Jacobi pseudoprime to base a. The smallest Euler-Jacobi pseudoprime to base 2 is 561.

  8. Fermat pseudoprime - Wikipedia

    en.wikipedia.org/wiki/Fermat_pseudoprime

    When p is a prime, p 2 is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the ...

  9. Mersenne prime - Wikipedia

    en.wikipedia.org/wiki/Mersenne_prime

    Since Mersenne numbers grow very rapidly, the search for Mersenne primes is a difficult task, even though there is a simple efficient test to determine whether a given Mersenne number is prime: the Lucas–Lehmer primality test (LLT), which makes it much easier to test the primality of Mersenne numbers than that of most other numbers of the ...