Proof of some conjectural hypergeometric supercongruences via curious identities

TL;DR

This paper proves several conjectured hypergeometric supercongruences involving binomial sums for primes, using novel hypergeometric identities, advancing understanding of supercongruences in number theory.

Contribution

The paper introduces new hypergeometric identities that enable the proof of longstanding conjectured supercongruences by Z.-W. Sun.

Findings

Proved supercongruence for sum involving binomial coefficients modulo p^2.

Derived explicit congruences depending on p mod 3.

Established new identities linking hypergeometric sums to binomial coefficients.

Abstract

In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{4k}{2k+1}\binom{2k}k}{48^k}\equiv0\pmod{p^2},$$ and $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv1\pmod{3},\\ p/\binom{(2p+2)/3}{(p+1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv2\pmod{3}.\end{cases} $$ We also obtain some other results of such types.