An analog of Wolstenholme's Theorem

Abstract

In this paper we shall prove an analogous version of Wolstenholme's theorem, namely, given a prime number p>=2 and positive integers a,b,m such that p|-m, we shall determine the maximal prime power p^e which divides the numerator of the fraction

1/m=1/(m+p^b)+...+1/(m+(p^a-1)p^b),

when written in reduced form, with the exception of one case, where p=2, b=1, m>1 and 2^a||m-1. In this exceptional case, a lower bound for e is given.