New Congruences on Multiple Harmonic Sums and Bernoulli Numbers (1504.03227v4)

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 $p \ge 11$ and integer $r\ge 2$, we prove that $$ \sum\limits_{\begin{smaLLMatrix} {{l}{1}}+{{l}{2}}+\cdots +{{l}{6}}={{p}^{r}} {{l}{1}},\cdots ,{{l}{6}}\in {\mathcal{P}{p}} \end{smaLLMatrix}}{\frac{1}{{{l}{1}}{{l}{2}}{{l}{3}}{{l}{4}}{{l}{5}}{l}{6}}}\equiv - \frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernoulli numbers.