Proof of two supercongruences conjectured by Z.-W. Sun
Number Theory
2021-11-18 v2 Combinatorics
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 , \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \end{align*} In fact, this supercongruence is a generalization of a supercongruence of van Hamme.
Cite
@article{arxiv.1910.00779,
title = {Proof of two supercongruences conjectured by Z.-W. Sun},
author = {Guo-Shuai Mao and Chen-Wei Wen},
journal= {arXiv preprint arXiv:1910.00779},
year = {2021}
}