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Wilson's Theorem, which states that , may be generalized to prime powers as follows:

This was proved by Gauss in * Disquisitiones Arithmeticae*,
as follows:\
If we pair up each **m** in the product with its inverse
then we 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 follows.

We may deduce from Lemma 1 the following

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

** Proof**: Write each **r** in the product below as , to get

In 1808 Legendre showed that the exact power of **p** dividing is

Let **r = n-m** and write each of
**n, m** and **r** in base **p**. Let if there is a `carry'
in the **j**th digit when adding **m** and **r** in base **p**, let
otherwise (including ).
We use (2.3) to deduce

** Kummer's Theorem.** * The power to which prime
p divides the binomial coefficient is given
by the number of `carries' when we add m and n-m in base p.*

** Proof**:
One can easily check that, for each integer ,

By a similar argument we also have, for each , that

This can be seen by letting be the least residues, in absolute value, of , respectively, so that times the left side of (2.4), plus equalsThe improvement, (1.2), of Lucas' Theorem is easily deduced from the equation

which was discovered by Anton, Stickelberger and then Hensel. For an arbitrary prime power , this may be generalized as follows:With definitions as in Theorem 1, we have, for any given ,

by Corollary 1. Multiplying together this congruence for each we get
** Proposition 1.** * For any integer n and prime power ,
we have
*

Theorem 1 then follows from dividing the equation in Proposition 1 by
the corresponding ones for **m** and **r**, and then using (2.4) to sort
out the exponents of and **p**.

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