# Extending Babbage's (Non-)Primality Tests

## TL;DR

This paper revisits Charles Babbage's early 19th-century primality test, extends it to identify the least prime factor, and explores related congruences, connecting historical and modern number theory insights.

## Contribution

It introduces an extension of Babbage's primality criterion to find the least prime factor and proves a partial converse, enriching the understanding of primality tests.

## Key findings

- Extended Babbage's primality test to least-prime-factor detection
- Proved a partial converse of Babbage's non-primality test
- Connected historical and modern number theory concepts

## Abstract

We recall Charles Babbage's 1819 criterion for primality, based on simultaneous congruences for binomial coefficients, and extend it to a least-prime-factor test. We also prove a partial converse of his non-primality test, based on a single congruence. Two problems are posed. Along the way we encounter Bachet, Bernoulli, Bezout, Euler, Fermat, Kummer, Lagrange, Lucas, Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary mathematicians.