An extension of Lucas Theorem

TL;DR

This paper extends Lucas Theorem by providing elementary proofs of congruence criteria for binomial coefficients modulo a prime, enabling new identities and congruences involving binomial sums.

Contribution

It introduces an extended version of Lucas Theorem using new congruence criteria and demonstrates their applications in deriving binomial coefficient identities.

Findings

Extended Lucas Theorem established with elementary proofs

New congruence criteria for binomial coefficients modulo a prime

Method for deriving identities and congruences involving binomial sums

Abstract

We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using those criteria. We give applications of these criteria by describing a method to derive identities and congruences involving sums of binomial coefficients.