MULTIPLE HARMONIC SUMS AND WOLSTENHOLME'S THEOREM

Abstract

We give a family of congruences for the binomial coefficient [Formula: see text], with k an integer and p a prime. Our congruences involve multiple harmonic sums, and hold modulo arbitrary large powers of p. The general congruence in our family, which depends on a parameter n, involves n "elementary symmetric" multiple harmonic sums, and holds modulo p2n+3. These congruences are actually part of a much larger collection of congruences for [Formula: see text] in terms of the elementary symmetric multiple harmonic sums. Congruences in our family have been optimized, in that they involve the fewest multiple harmonic sums among those congruences holding modulo the same power of p. The coefficients in our congruences are given by polynomials in k.