Squares in arithmetic progressions and infinitely many primes

TL;DR

This paper presents a novel proof of the infinitude of primes using combinatorial and number-theoretic ideas, connecting van der Waerden's theorem and properties of squares in arithmetic progressions.

Contribution

It introduces a new proof of infinitely many primes based on combinatorial coloring and Fermat's theorem about squares in arithmetic progressions.

Findings

New proof of infinitely many primes

Connection between combinatorics and number theory

Discussion of historical intersections of these ideas

Abstract

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.