Some congruences related to a congruence of Van Hamme

TL;DR

This paper proves new supercongruences related to Van Hamme's conjecture using advanced combinatorial and number-theoretic techniques, and introduces two related conjectures.

Contribution

It establishes novel supercongruences involving Euler numbers and hypergeometric sums, employing the Wilf--Zeilberger method, Whipple's transformation, and computational tools.

Findings

Proved supercongruence involving Euler numbers and hypergeometric sums.

Used Wilf--Zeilberger method and software Sigma for proofs.

Proposed two new related conjectures.

Abstract

We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\ \sum_{k=0}^{(p+1)/2} (4k-1)^5 \frac{(-\frac{1}{2})_k^4}{k!^4} &\equiv 16p\pmod{p^{4}}, \end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\tt Sigma} developed by Schneider. We also put forward two related conjectures.