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