Proof of two supercongruences of truncated hypergeometric series ${}_4{F}_3$

Abstract

In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{6n+1}{(-512)^n}\binom{2n}n^3&\equiv p\left(\frac{-2}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\pmod{p^4}, \end{align*} where $\left(\frac{\cdot }{p}\right)$ stands for the Legendre symbol, and $E_n$ is the $n$-th Euler number.

Accepted by Acta Mathematica Sinica, English Series

References in corpus (2)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• Proof of two supercongruences by the Wilf-Zeilberger method