Fermat's little theorem

Fermat's little theorem states that if p is a prime number, then for any integer a,

<math>a^p = a \pmod{p}</math>

This means that if you you take some number a, multiply it by itself p times and subtract a, the result is divisible by p (see modular arithmetic). It is called Fermat's little theorem to differentiate it from Fermat's last theorem. Pierre de Fermat found the theorem around 1636; it appeared in one of his letters, dated October 18, 1640 to his confidante Frenicle in the following equivalent form: p divides a^p-1 - 1 whenever p is prime and a is coprime to p. The case a = 2 was known to the ancient Chinese.

Proofs

Fermat explained his theorem without a proof. The first one who gave a proof was Gottfried Wilhelm Leibniz in a manuscipt without a date, where he wrote also that he knew a proof before 1683.

See Proofs of Fermat's little theorem.

Generalizations

A slight generalization of the theorem, which immediately follows from it, is as follows: if p is prime and m and n are positive integers with m ≡ n (mod p-1), then a^m ≡ aⁿ (mod p) for all integers a. In this form, the theorem is used to justify the RSA public key encryption method.

Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have

<math>a^{\varphi (n)} = 1 \pmod{n}</math>

where φ(n) denotes Euler's φ function counting the integers between 1 and n that are coprime to n. This is indeed a generalization, because if n = p is a prime number, then φ(p) = p - 1.

This can be further generalized to Carmichael's theorem[?], stated here: http://mathworld.wolfram.com/CarmichaelsTheorem.

Pseudoprimes

If a and p are coprime numbers such that a^p-1 - 1 is divisible by p, then p need not be prime. If it is not, then p is called a pseudoprime to base a. A number p that is a pseudoprime to base a for every number a coprime to p is called a Carmichael number.