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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 ).
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.}
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 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 ...
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}}.}
Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. [3] The representation of each such number is unique, up to the ordering of the two squares. [4]
For integer b > 1, base b may be used if and only if only a finite number of Fermat numbers F n satisfies that () =, where () is the Jacobi symbol. In fact, Pépin's test is the same as the Euler-Jacobi test for Fermat numbers, since the Jacobi symbol ( b F n ) {\displaystyle \left({\frac {b}{F_{n}}}\right)} is −1, i.e. there are no Fermat ...
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 ...