Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Proof of two supercongruences conjectured by Z.-W.Sun involving
  Catalan-Larcombe-French numbers   The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play
important roles in mathematics. With helps of some combinatorial identities, we
establish the following two congruences:
$$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\
\mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\
\mbox{modulo}\ p$$ for any prime $p>3$, the second one was conjectured by Z.-W.
Sun in 2012. These two congruences are very important to prove the following
conjectures of Z.W.Sun: For any old prime $p$, we have
$$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\big(\frac{-1}p\big)p^2E_{p-3}\pmod{p^3}$$
and
$$\sum_{k=0}^{p-1}\frac{P_k}{{16}^k}\equiv\big(\frac{-1}p\big)-p^2E_{p-3}\pmod{p^3},$$
where $P_n=\sum_{k=0}^n\frac{\binom{2k}k^2\binom{2(n-k)}{n-k}^2}{\binom nk}$ is
the n-th Catalan-Larcombe-French number.

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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat