![]() ![]() ![]() This result may be deduced from Fermat's little theorem by the fact that, if p is an odd prime, then the integers modulo p form a finite field, in which 1 has exactly two square roots, 1 and −1. If p is an odd prime, and p – 1 = 2 s d with d odd, then for every a prime to p, either a d ≡ 1 mod p, or there exists t such that 0 ≤ t < s and a 2 t d ≡ −1 mod p. The Miller–Rabin primality test uses the following extension of Fermat's little theorem: Is either a prime or a Carmichael number. In the notation of modular arithmetic, this is expressed asĪ p ≡ a ( mod p ). For other theorems named after Pierre de Fermat, see Fermat's theorem.įermat'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.
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