# On two conjectured supercongruences involving truncated hypergeometric
series

Research paper by **Guo-Shuai Mao, Hao Pan**

Indexed on: **07 Jan '18**Published on: **07 Jan '18**Published in: **arXiv - Mathematics - Number Theory**

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#### Abstract

In this paper, we prove two conjectured supercongruences of Sun. For example,
let $p$ be an odd prime and $r\geq 1$. Suppose that $x$ is a $p$-adic integer
with $x\equiv-2k\pmod p$ for some $1\leq k\leq (p+1)/(2r+1)$. Then we have
$${}_{2r+1}F_{2r}\bigg[\begin{matrix}-x&-x&\ldots&-x\\
&1&\ldots&1\end{matrix}\bigg|\,1\bigg]_{p-1}\equiv0\pmod{p^2},$$ where $$
{}_{q+1}F_{q}\bigg[\begin{matrix}x_0&x_1&\ldots&x_{q}\\
&y_1&\ldots&y_q\end{matrix}\bigg|\,z\bigg]_{n}=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_q)_k}{(y_1)_k\cdot
(y_q)_k}\cdot\frac{z^k}{k!}. $$