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
![](Image/img135.gif)
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
![](Image/img135a.gif)
where
is as in Lemma 1.
Proof:
We write each r in the product below as
, to get
(20)![](Image/equation20.gif)
by Lemma 1, where
signifies a product over integers not divisible by p.