Proof of a conjectural supercongruence

TL;DR

This paper proves a supercongruence involving binomial coefficients and alternating signs, confirming a conjecture of Sun for primes greater than a certain bound, advancing understanding in number theory.

Contribution

The paper establishes a new supercongruence involving binomial coefficients, confirming Sun's conjecture for primes exceeding a specific threshold.

Findings

Supercongruence holds modulo p^3 for primes p > m q.

Confirms Sun's conjecture in number theory.

Provides new insights into binomial coefficient congruences.

Abstract

Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the supercongruence $$\sum_{k=0}^{p-1}(-1)^{km}\binom{p/m-q}{k}^m\equiv0\pmod{p^3}.$$ for any prime $p>mq$. This confirms a conjecture of Sun.