Fermat's Little Theorem
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
For example, if a = 2 and p = 7, 27 = 128, and 128 − 2 = 7 × 18 is an integer multiple of 7.
If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols
For example, if a = 2 and p = 7 then 26 = 64 and 64 − 1 = 63 is thus 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.
Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following:
If p is a prime and a is any integer not divisible by p, then a p − 1 − 1 is divisible by p.
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