New Congruences on Multiple 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 $p\ge 11$ and integer $r\ge 2$, we prove that $$\sum _{\begin{array}{c}{l}_1+l_2+\cdots +l_6=p^r{l}_1,\cdots ,l_6\in {\mathcal{P}}_p\end{array}}\frac{1}{l_1{l}_2{l}_3{l}_4{l}_5{l}_6}\equiv -\frac{5!}{18}{p}^{r-1}{B}_{p-3}^2(\mathrm{mod}{p}^r).$$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernoulli numbers.

14 pages. This version simplifies the previous version. Moreover, two important theorems and a conjecture were added

References in corpus (4)

• A congruence involving harmonic sums modulo $p^{\alpha }{q}^{\beta }$
• A Curious Congruence Involving Alternating Harmonic Sums
• Super congruences involving alternating harmonic sums modulo prime powers
• A congruence involving alternating harmonic sums modulo $p^{\alpha }{q}^{\beta }$

Cited by in corpus (2)

• Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values
• Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums