Fermat's little theorem

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 $${\displaystyle a^{p}\equiv a{\pmod {p}}.}$$

For example, if a = 2 and p = 7, then 2⁷ = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7.

If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a^{p − 1} − 1 is an integer multiple of p, or in symbols:^[1][2] $${\displaystyle a^{p-1}\equiv 1{\pmod {p}}.}$$

For example, if a = 2 and p = 7, then 2⁶ = 64, and 64 − 1 = 63 = 7 × 9 is a multiple of 7.

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.^[3]