# 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*.

** Proof:**

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
**(20)**

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