On two conjectural supercongruences of Apagodu and Zeilberger

TL;DR

This paper proves two conjectured supercongruences involving sums of binomial coefficients and their relations modulo prime squares, extending understanding of number theory and combinatorial identities.

Contribution

It establishes the validity of two supercongruences conjectured by Apagodu and Zeilberger, providing proofs for their relations involving binomial sums and prime modulus.

Findings

Proves lpha_{np} \u2261 (rac{p}{3}) \u00d7 lpha_n mod p^2.

Establishes eta_{np} \u2261 eta_n mod p^2 for p 2 1 mod 3.

Shows eta_{np} \u2261 -_n mod p^2 for p 2 2 mod 3.

Abstract

Let the numbers $\alpha_n,\beta_n$ and $\gamma_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime $p\ge 5$ and positive integer $n$ \begin{align*} \alpha_{np}&\equiv \left(\frac{p}{3}\right) \alpha_n \pmod{p^2},\\ \beta_{np}&\equiv \begin{cases} \displaystyle \beta_n \pmod{p^2},\quad &\text{if $p\equiv 1\pmod{3}$},\\ -\gamma_n \pmod{p^2},\quad &\text{if $p\equiv 2\pmod{3}$}, \end{cases} \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol. These two supercongruences were recently conjectured by Apagodu and Zeilberger.