Some new curious congruences involving multiple harmonic sums

Rong Ma, Ni Li

Published 2023-05-13Version 1

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been generalized along different directions, where $B_n$ denote the $n$th Bernoulli number. In this paper, we obtain several new generalizations of the above congruence by applying congruences involving multiple harmonic sums. For example, we have $$\sum_{\substack{k_1+k_2+\cdots+k_n=p \\ k_i> 0, 1 \le i \le n}} \dfrac{(-1)^{k_1}\left(\dfrac{k_1}{3}\right)}{k_1 \cdots k_n} \equiv \dfrac{(n-1)!}{n}\dfrac{2^{n-1}+1}{3\cdot6^{n-1}}B_{p-n}\left(\dfrac{1}{3}\right)\pmod p,$$ where $n$ is even, $B_n(x)$ denote the Bernoulli polynomials.