Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proof of some congruence conjectures of Guo and Liu   Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by
$S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$ In this paper we
prove some conjectures of Guo and Liu which extend some conjectures of Z.-W.
Sun \cite{Su1}, such as: There exist integers $a_{2r-1}$ and $b_r$, independent
of $n$, such that $$a_{2r-1}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv0\pmod{n^2}\
\mbox{and}\ b_r\sum_{k=0}^{n-1}kS_k^{(r)}\equiv0\pmod{n^2}.$$ By Zeilberger
algorithm, we find that for all $0\leq j<n$,
$$(2j+1)\binom{2j}j\sum_{k=j}^{n-1}(2k-j+1)\binom kj^2\equiv0\pmod{n^2}.$$

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The categories of this paper are: eess
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin