Supercongruences via Beukers' method

TL;DR

This paper employs Beukers' method to prove various supercongruences involving binomial coefficients, Apéry-like numbers, and related sequences modulo prime powers, extending the understanding of these congruences in number theory.

Contribution

The paper applies Beukers' unified method to establish new supercongruences for complex binomial sums and Apéry-like numbers, confirming several conjectures in the field.

Findings

Proved supercongruences for binomial coefficient sums modulo p^3.

Confirmed conjectures related to Apéry-like numbers and their congruences.

Extended Beukers' method to new classes of supercongruences.

Abstract

Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Ap\'ery number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.