A proof of Wilson's Theorem

By Chris Caldwell

In 1770 Edward Waring announced the following theorem by his former student John Wilson.

Wilson's Theorem.

Let p be an integer greater than one. p is prime if and only if (p-1)! ≡ -1 (mod p).

This beautiful result is of mostly theoretical value because it is relatively difficult to calculate (p-1)! In contrast it is easy to calculate a^p-1, so elementary primality tests are built using Fermat's Little Theorem rather than Wilson's.

Neither Waring or Wilson could prove the above theorem, but now it can be found in any elementary number theory text. To save you some time we present a proof here.

Proof.

It is easy to check the result when p is 2 or 3, so let us assume p > 3. If p is composite, then its positive divisors are among the integers

1, 2, 3, 4, ... , p-1

and it is clear that gcd((p-1)!,p) > 1, so we can not have (p-1)! ≡ -1 (mod p).

However if p is prime, then each of the above integers are relatively prime to p. So for each of these integers a there is another b such that ab ≡ 1 (mod p). It is important to note that this b is unique modulo p, and that since p is prime, a ≡ b if and only if a is 1 or p-1. Now if we omit 1 and p-1, then the others can be grouped into pairs whose product is one showing

2^.3^.4^....^.(p-2) ≡ 1     (mod p)

(or more simply (p-2)! ≡ 1 (mod p)). Finally, multiply this equality by p-1 to complete the proof. ∎