A LUCAS TYPE THEOREM MODULO PRIME POWERS

In this note we prove that (np(s) mp(s)+r) equivalent to (-1)(r-1)r(-1) (m + 1) (n m+1)p(s) (mod p(s+1)) where p is any prime, n, m, s and r are nonnegative integers such that n >= m, s >= 1, 1 <= r <= p(s) - 1 and r is not divisible by p. We derive a proof by induction using a multiple application of Lucas' Theorem and two basic binomial coefficient identities. As an application, we prove that a similar congruence for a prime p >= 5 established in 1992 by D. F. Bailey holds for all primes p.