A Generalization of Lucas' Theorem.

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 jth digit, when adding m and r in base p.

Theorem 1. For any prime power and positive integers m=n+r, we have


where is except if p=2 and .

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



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.