Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the supercongruences involving harmonic numbers of order 2   We prove several supercongruences involving the harmonic number of order two
$H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $\alpha$ is
$p$-integral, then we can completely determine $$
\sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k}\quad\text{and}\quad
\sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k}
$$ modulo $p^3$. In particular, by setting $\alpha=-1/2$, we confirm two
conjectured congruences of Z.-W. Sun.

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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas