Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New Congruences on Multiple Harmonic Sums and Bernoulli Numbers   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 $$ \sum\limits_{\begin{smallmatrix}
  {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{6}}={{p}^{r}}
  {{l}_{1}},\cdots ,{{l}_{6}}\in {\mathcal{P}_{p}}
\end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\equiv
- \frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \pmod{{{p}^{r}}}. $$ This extends a family
of curious congruences. We also obtain other interesting congruences involving
multiple harmonic sums and Bernoulli numbers.

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The categories of this paper are: patt-sol
18
The categories of this paper are: econ
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The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg