A New Super Congruence Involving Multiple Harmonic Sums (1410.1712v2)

Abstract: Let ${\mathcal{P}{n}}$ denote the set of positive integers which are prime to $n$. Let $B{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smaLLMatrix} {{l}{1}}+{{l}{2}}+\cdots +{{l}{5}}={{p}^{r}} {{l}{1}},\cdots ,{{l}{5}}\in {\mathcal{P}{p}} \end{smaLLMatrix}}{\frac{1}{{{l}{1}}{{l}{2}}{{l}{3}}{{l}{4}}{{l}{5}}}}\equiv -\frac{5!}{6}{{B}{p-5}}{{p}^{r-1}} \pmod{{{p}^{r}}}. \end{equation} This gives an extension of a family of super congruences found by Wang, Cai and Zhao.