A primality criterion based on a Lucas' congruence

TL;DR

This paper generalizes Lucas' primality criterion by showing that if a binomial coefficient congruence holds for all k, then the number involved must be a prime power, providing a new characterization of primes.

Contribution

It extends Lucas' congruence to a broader setting, establishing that such congruences imply the number is a prime power and the modulus is prime.

Findings

If the binomial congruence holds for all k, then q is prime.

Under the same conditions, n must be a power of q.

The result characterizes prime powers via binomial coefficient congruences.

Abstract

Let $p$ be a prime. In 1878 \'{E}. Lucas proved that the congruence $$ {p-1\choose k}\equiv (-1)^k\pmod{p}$$ holds for any nonnegative integer $k\in\{0,1,\ldots,p-1\}$. The converse statement was given in Problem 1494 of {\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In this note we generalize this converse assertion by the following result: If $n>1$ and $q>1$ are integers such that $$ {n-1\choose k}\equiv (-1)^k \pmod{q}$$ for every integer $k\in\{0,1,\ldots, n-1\}$, then $q$ is a prime and $n$ is a power of $q$.