Symbolic summation methods and hypergeometric supercongruences

TL;DR

This paper proves new hypergeometric supercongruences involving sums with binomial coefficients and Euler numbers, confirming a recent conjecture and advancing the understanding of symbolic summation in number theory.

Contribution

The paper establishes two new supercongruences involving hypergeometric sums and Euler numbers, confirming a conjecture by Guo and Schlosser.

Findings

Proved a supercongruence modulo p^4 involving hypergeometric sums.

Confirmed a conjecture about congruences modulo p^3 for specific sums.

Connected hypergeometric identities with Euler numbers in number theory.

Abstract

In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.