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arXiv · 1910.09983

Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$

Guo-Shuai Mao
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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.

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BibTeXRIS

Guo-Shuai Mao. 2019-10-20. Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$. https://doi.org/10.1007/s10114-023-1152-8

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