Proof of a conjectural supercongruence modulo $p^5$

TL;DR

This paper proves a complex supercongruence involving binomial coefficients and Bernoulli numbers modulo p^5, confirming a conjecture by Sun from 2019.

Contribution

It provides a rigorous proof of Sun's conjectured supercongruence, advancing understanding of p-adic properties of binomial sums.

Findings

Confirmed the supercongruence for all primes p > 3

Established a link between binomial sums and Bernoulli numbers modulo p^5

Validated a conjecture that was open since 2019

Abstract

In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.