Super congruences involving alternating harmonic sums modulo prime powers

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} (\bmod p^{r}), \end{equation*} where $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we establish a combinational congruence of alternating harmonic sums for any odd prime $p$ and positive integers $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}p^{r-1}B_{p-3} (\bmod p^{r}). \end{equation*} For any odd prime $p\geq 5$ and positive integers $r$, we have \begin{align} &4\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}}+3\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}} \nonumber\\&\equiv\begin{cases} \frac{216}{5}pB_{p-5}\pmod{p^{2}}, if r=1, \\ \frac{36}{5}p^{r}B_{p-5}\pmod{p^{r+1}}, if r>1. \\ \end{cases}\nonumber \end{align} For any odd prime $p> 5$ and positive integers $r$, we have \begin{align} &\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}i_{5}}+2\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}i_{5}} \nonumber\\&\equiv\begin{cases} 12B_{p-5}\pmod{p}, if r=1,\\ 6p^{r-1}B_{p-5}\pmod{p^{r}}, if r>1. \end{cases}\nonumber \end{align}