# $p$-adic supercongruences conjectured by Sun

Research paper by **Yong Zhang**

Indexed on: **06 May '21**Published on: **31 Oct '19**Published in: **arXiv - Mathematics - Combinatorics**

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#### Abstract

In this paper we prove three results conjectured by Z.-W. Sun. Let $p$ be an
odd prime and let $h\in \mathbb{Z}$ with $2h-1\equiv0\pmod{p^{}}$. For
$a\in\mathbb{Z}^{+}$ and $p^a>3$, we show that \begin{align}\notag
\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}\bigg(-\frac{h}{2}\bigg)^k\equiv0\pmod{p^{a+1}}.
\end{align} Also, for any $n\in \mathbb{Z}^{+}$ we have \begin{align} \notag
\nu_{p}\bigg(\sum_{k=0}^{n-1}\binom{hn-1}{k}\binom{2k}{k}\bigg(-\frac{h}{2}\bigg)^k\bigg)\geq\nu_{p}(n)\notag,
\end{align} where $\nu_p(n)$ denotes the $p$-adic order of $n$. For any integer
$m\not\equiv 0\pmod{p^{}}$ and positive integer $n$, we have \begin{align*}
\frac{1}{pn}\bigg(\sum_{k=0}^{pn-1}\binom{pn-1}{k}\frac{\binom{2k}{k}}{(-m)^k}-\bigg(\frac{m(m-4)}{p}\bigg)\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{\binom{2k}{k}}{(-m)^k}\bigg)\in
\mathbb{Z}_{p}, \end{align*} where $(\frac{.}{})$ is the Legendre symbol and
$\mathbb{Z}_p$ is the ring of $p$-adic integers.