A characterization of a prime $p$ from the binomial coefficient ${n \choose p}$

TL;DR

This paper completes a proof of a theorem characterizing prime numbers using binomial coefficients and extends it with a related generalized result, inspired by an Olympiad problem.

Contribution

It provides a complete proof of a prime characterization theorem involving binomial coefficients and introduces a generalization of this theorem.

Findings

Prime characterization via binomial coefficients

Complete proof of the theorem

Generalization of the prime characterization

Abstract

We complete a proof of a theorem that was inspired by an Indian Olympiad problem, which gives an interesting characterization of a prime number $p$ with respect to the binomial coefficients ${n\choose p}$. We also derive a related result which generalizes the theorem in one direction.