If p is a prime number and a a natural number, then

(1)

Furthermore, if (p does not divide a), then there exists some smallest exponent d such that

(2)

and d divides . Hence,

(3)

The theorem is sometimes also simply known as "Fermat's theorem" (Hardy and Wright 1979, p. 63).

This is a generalization of the Chinese hypothesis and a special case of Euler's 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 MacLane (1940) and Kaplansky (1945).

The theorem is easily proved using mathematical induction on a. Suppose (i.e., p divides ). Then examine

(4)

From the binomial theorem,

(5)

Rewriting,

(6)

But p divides the right side, so it also divides the left side. Combining with the induction hypothesis gives that p divides the sum

(7)

as assumed, so the hypothesis is true for any a. 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 p is prime, there does not exist a base a < p with such that possesses a nonzero residue modulo p. If such base a exists, p is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat's little theorem does not guarantee that p 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 as 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 a relatively prime to p. However, Fermat's little theorem converse provides a criterion for certifying the primality of a number. A table of the smallest pseudoprimes P for the first 100 bases a follows (Sloane's A007535; Beiler 1966, p. 42 with typos corrected).

Binomial Theorem, Carmichael Number, Chinese Hypothesis, Composite Number, Compositeness Test, Euler's Theorem, Fermat's Little Theorem Converse, Fermat Pseudoprime, Modulo Multiplication Group, Pratt Certificate, Primality Test, Prime Number, Pseudoprime, Relatively Prime, Totient Function, Wieferich Prime, Wilson's Theorem, Witness

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