Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Super congruences involving alternating harmonic sums modulo prime
  powers   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}

8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th