On binomial coefficients modulo squares of primes

TL;DR

This paper provides elementary proofs for several binomial coefficient congruences modulo the square of primes, extending known results with simplified methods and explicit formulas involving prime residue classes.

Contribution

It introduces elementary proofs for complex binomial coefficient congruences modulo prime squares, simplifying previous approaches and clarifying the role of prime residue classes.

Findings

Proves sum of binomial coefficients modulo p^2 depends on p mod 3.

Establishes congruences for sums involving binomial coefficients with scaled indices.

Provides explicit formulas for binomial sums based on prime residue classes.

Abstract

We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum\limits_{n=0}^{p-1}\dbinom{2n}{n} \equiv\eta_{p}\mod p^{2},$$ $$\sum\limits_{n=0}^{rp-1}\dbinom{2n}{n} \equiv\eta_{p}\sum\limits_{n=0}^{r-1}\dbinom {2n}{n}\mod p^{2}$$ and $$\sum\limits_{n=0}^{rp-1}\sum\limits_{m=0}^{sp-1}\dbinom{n+m}{m}^{2} \equiv\eta_{p} \sum\limits_{m=0}^{r-1}\sum\limits_{n=0}^{s-1}\dbinom{n+m}{m}^2\mod p^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $\eta_{p}= \begin{cases} 0, &\text{if }p\equiv0\mod 3;\\ 1, & \text{if }p\equiv1\mod 3;\\ -1, &\text{if }p\equiv2\mod 3 \end{cases}.$$