Note on super congruences modulo $p^2$

TL;DR

This paper investigates supercongruences involving binomial coefficient sums modulo p^2, extending known congruences modulo p for primes p and specific parameters.

Contribution

It establishes a new supercongruence result showing that a sum congruent to zero modulo p also holds modulo p^2 under certain conditions.

Findings

Proves that certain binomial sum congruences modulo p extend to modulo p^2.

Provides conditions under which the congruence modulo p implies the same modulo p^2.

Enhances understanding of supercongruences in number theory.

Abstract

Let $p$ be an odd prime, and let $m$ be an integer with $p\nmid m$. In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hbox{implies}\quad\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak \binom{-1-a}k}{m^k}\equiv 0\pmod {p^2}.$$