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}}).]