Proof of two supercongruences by the Wilf-Zeilberger method

TL;DR

This paper proves two supercongruences using the Wilf-Zeilberger method, including confirming a conjecture of Sun involving prime numbers, Euler polynomials, and binomial coefficients.

Contribution

It introduces a novel application of the Wilf-Zeilberger method to prove supercongruences, confirming a specific conjecture of Sun.

Findings

Proved a supercongruence involving primes and Euler polynomials.

Confirmed Sun's conjecture for the case n=1.

Demonstrated the effectiveness of the Wilf-Zeilberger method in number theory.

Abstract

In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}n^3\equiv p\left(\frac{-1}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\left(\frac14\right)\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}(x)$ are the Euler polynomials. This congruence confirms a conjecture of Sun \cite[(2.18)]{sun-numb-2019} with $n=1$.