On a congruence involving harmonic series and Bernoulli numbers

Abstract

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime p [Formula: see text] where [Formula: see text] is the nth Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime p in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer n with an odd prime factor p such that [Formula: see text] where [Formula: see text] [Formula: see text] In this paper, we establish the following generalization of their conjecture: for any positive integer n with an odd prime factor [Formula: see text] such that [Formula: see text] where [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are positive integers coprime to [Formula: see text], and [Formula: see text] is a positive common multiple of [Formula: see text], [Formula: see text] and [Formula: see text]. Also, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].