Proof of two supercongruences conjectured by Z.-W. Sun

TL;DR

This paper proves two supercongruences conjectured by Z.-W. Sun using the Wilf-Zeilberger method, extending known results and providing new insights into supercongruence properties related to binomial coefficients and primes.

Contribution

The paper introduces a proof of two supercongruences conjectured by Sun, including a generalization of van Hamme's supercongruence, using the Wilf-Zeilberger method.

Findings

Proved two supercongruences conjectured by Sun.

Extended a supercongruence of van Hamme.

Validated supercongruence involving binomial coefficients and primes.

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}\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.