For a given integer *n* define to be the product of those integers
that are not divisible by *p*. We write in base *p*,
and let be the least positive residue of
for each , so that . Finally we
let be the number of
`carries' on or beyond the *j*th digit, when adding *m* and *r* in base *p*.

*
where is except if p=2 and . *

Now, with definitions as in Theorem 1, we have, for any given ,

by

**(20)**.

Multiplying together this congruence for each we get

** Proposition 1. ** * For any integer n and prime power ,
we have
*

*
where is defined as in Lemma 1 and each as in Theorem 1.*

Theorem 1 then follows from dividing the equation in Proposition 1 by
the corresponding ones for *m* and *r*, and then using (19) to sort
out the exponents of and *p*.