A Curious Congruence Involving Alternating Harmonic Sums

Abstract

Let $p$ be a prime and ${\mathcal{P}}_p$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{{{(-1)}^{i}}}{ijk}}\equiv \frac{{{p}^{r-1}}}{2}{{B}_{p-3}}\, (\bmod \, {{p}^{r}}).\]

This is an original research article about congruences. It has submitted for publication in June 2014

Cited by in corpus (1)

• Super congruences involving alternating harmonic sums modulo prime powers