Proof of Fermat's Little Theorem

By Chris Caldwell

Fermat's "biggest", and also his "last" theorem states that xⁿ + yⁿ = zⁿ has no solutions in positive integers x, y, z with n > 2. This has finally been proven by Wiles in 1995. Here we are concerned with his "little" but perhaps his most used theorem which he stated in a letter to Fre'nicle on 18 October 1640:

Fermat's Little Theorem.

Let p be a prime which does not divide the integer a, then a^p-1 ≡ 1 (mod p).

It is so easy to calculate a^p-1 quickly modulo p that most elementary primality tests are built using a version of Fermat's Little Theorem rather than Wilson's Theorem.

As usual Fermat did not provide a proof (this time saying "I would send you the demonstration, if I did not fear its being too long" [Burton80, p79]). Euler first published a proof in 1736, but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.

Proof.

Start by listing the first p-1 positive multiples of a:

a, 2a, 3a, ... (p -1)a

Suppose that ra and sa are the same modulo p, then we have r = s (mod p), so the p-1 multiples of a above are distinct and nonzero; that is, they must be congruent to 1, 2, 3, ..., p-1 in some order. Multiply all these congruences together and we find

a (2a) (3a) ... ((p-1)a) ≡ 1^.2^.3^....^.(p-1) (mod p)

which is, a^(p-1)(p-1)! ≡ (p-1)! (mod p).   Divide both side by (p-1)! to complete the proof. ∎

Sometimes Fermat's Little Theorem is presented in the following form:

Corollary.

Let p be a prime and a any integer, then a^p ≡ a (mod p).

Proof.

The result is trival (both sides are zero) if p divides a. If p does not divide a, then we need only multiply the congruence in Fermat's Little Theorem by a to complete the proof. ∎