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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 ).
For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
The multiplicative property of the norm implies that a prime number p is either a Gaussian prime or the norm of a Gaussian prime. Fermat's theorem asserts that the first case occurs when p = 4 k + 3 , {\displaystyle p=4k+3,} and that the second case occurs when p = 4 k + 1 {\displaystyle p=4k+1} and p = 2. {\displaystyle p=2.}
The probability of the existence of another Fermat prime is less than one in a billion. ... Primes p that divide 2 n − 1, for some prime number n. 3, 7, 23, 31, 47 ...
A Mersenne–Fermat number is defined as 2 p r − 1 / 2 p r − 1 − 1 with p prime, r natural number, and can be written as MF(p, r). When r = 1 , it is a Mersenne number. When p = 2 , it is a Fermat number .
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic , this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.}
The false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2 340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2.
65537 is commonly used as a public exponent in the RSA cryptosystem. Because it is the Fermat number F n = 2 2 n + 1 with n = 4, the common shorthand is "F 4" or "F4". [4] This value was used in RSA mainly for historical reasons; early raw RSA implementations (without proper padding) were vulnerable to very small exponents, while use of high exponents was computationally expensive with no ...