If is a prime number and is a natural number, then
Furthermore, if ( does not divide ), then there exists some smallest exponent such that
and divides . Hence,
The theorem is sometimes also simply known as "Fermat'stheorem" (Hardy and Wright 1979, p. 63).
This is a generalization of the Chinese hypothesis and a special case of Euler's totient theorem. It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality. Although it was presumably proved (but suppressed) by Fermat, the first proof was published by Euler in 1749. It is unclear when the term "Fermat's little theorem" was first used to describe the theorem, but it was used in a German textbook by Hensel (1913) and appears in Mac Lane (1940) and Kaplansky (1945).
The theorem is easily proved using mathematical induction on . Suppose (i.e., divides ). Then examine
From the binomial theorem,
But divides the right side, so it also divides the left side. Combining with the induction hypothesis gives that divides the sum
as assumed, so the hypothesis is true for any . The theorem is sometimes called Fermat's simple theorem. Wilson's theorem follows as a corollary of Fermat's little theorem.
Fermat's little theorem shows that, if is prime, there does not exist a base with such that possesses a nonzero residue modulo . If such base exists, is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat's little theorem does not guarantee that is prime. The property of unambiguously certifying composite numbers while passing some primes make Fermat's little theorem a compositeness test which is sometimes called the Fermat compositeness test. A number satisfying Fermat's little theorem for some nontrivial base and which is not known to be composite is called a probable prime.
Composite numbers known as Fermat pseudoprimes (or sometimes simply "pseudoprimes") have zero residue for some s and so are not identified as composite. Worse still, there exist numbers known as Carmichael numbers (the smallest of which is 561) which give zero residue for any choice of the base relatively prime to . However, Fermat's little theorem converse provides a criterion for certifying the primality of a number. A table of the smallest pseudoprimes for the first 100 bases follows (OEIS A007535; Beiler 1966, p. 42 with typos corrected).