Some Wolstenholme type congruences

Abstract

In this paper we give an extension and another proof of the following Wolstenholme's type curious congruence established in 2008 by J. Zhao. Let s and l be two positive integers and let p be a prime such that p ls + 3. Then H(fsgl; p����1) S(fsgl; p����1) 8>>< >>: ���� s(ls + 1)p2 2(ls + 2) Bp����ls����2 (mod p3) if 2 - ls (����1)l����1 sp ls + 1 Bp����ls����1 (mod p2) if 2 j ls: APs an application, for given prime p 5, we obtain explicit formulae for the sum 1 k1< <kl p����1 1=(k1 kl) (mod p3) if k 2 f1; 3; : : : ; p ���� 2g, and for the sum P 1 k1< <kl p����1 1=(k1 kl) (mod p2) if k 2 f2; 4; : : : ; p ���� 3g