Wilson's Theorem for prime powers

(The Proof of Lemma 1 and its Corollary)

Wilson's Theorem, states that . We generalize it, to prime powers, as follows:

Lemma 1 For any given prime power we have

where is -1, unless in which case is 1.

Pair up each m in the product with its inverse to see that is congruent, modulo , to the product of those that are not distinct from their inverses ; that is those m for which . It is easy to show that the only such m are 1 and unless (when one only has m=1) or when one has the additional solutions and . The result then follows.

Corollary 1 For any given prime power , let be the least non-negative residue of . Then

where is as in Lemma 1.

Proof: We write each r in the product below as , to get


by Lemma 1, where signifies a product over integers not divisible by p.