A Generalized Supercongruence of Z.-W. Sun

TL;DR

This paper proves a supercongruence conjecture involving binomial coefficients and primes using the Wilf-Zeilberger method, extending the understanding of congruences in number theory.

Contribution

It introduces a novel proof of a supercongruence conjecture by Z.-W. Sun employing WZ techniques and combinatorial identities, broadening supercongruence applications.

Findings

Confirmed the supercongruence conjecture for all primes p

Developed a proof technique using WZ and combinatorial identities

Extended supercongruence results to a generalized form

Abstract

In this paper, we employ the Wilf-Zeilberger (WZ) method to prove a supercongruence conjecture posed by Z.-W. Sun: for any prime $p$, \begin{align*} \sum_{k=0}^{\frac{p-3}{2}}\frac{92k^2+61k+9}{(2k+1)64^k}{2k \choose k}{3k \choose k}{4k \choose 2k}\equiv 6p+16p^2\left(\frac{-1}{p}\right) \pmod{p^3}, \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol. Our proof relies on combinatorial identities and symbolic summation techniques.