A New Super Congruence Involving Multiple Harmonic Sums

TL;DR

This paper establishes a new super congruence involving multiple harmonic sums and Bernoulli numbers, extending previous results and providing a deeper understanding of harmonic sum congruences modulo prime powers.

Contribution

The paper proves a novel super congruence for sums over integers coprime to a prime, extending earlier work by Wang, Cai, and Zhao on harmonic sum congruences.

Findings

Proves a super congruence involving multiple harmonic sums and Bernoulli numbers.

Extends known super congruences to higher prime powers.

Provides a new link between harmonic sums and Bernoulli numbers.

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.