Supercongruences involving products of two binomial coefficients modulo $p^4$

TL;DR

This paper proves a conjecture by Z.-W. Sun involving a supercongruence for a sum of binomial coefficient products modulo p^4, revealing deep properties of binomial sums and harmonic numbers in number theory.

Contribution

It establishes a new supercongruence involving binomial coefficients and harmonic numbers, confirming a conjecture by Sun and advancing understanding of binomial sum congruences.

Findings

Proved a supercongruence for a binomial sum modulo p^4.

Connected binomial coefficient sums with harmonic numbers in a new way.

Confirmed a conjecture by Z.-W. Sun on binomial sum supercongruences.

Abstract

In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic number.