Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the Infinitude of the Primes

TL;DR

This paper presents a new proof that the primes are infinite by leveraging Schur's Theorem from Ramsey Theory and Fermat's Last Theorem, offering a simpler approach than previous proofs.

Contribution

It introduces a novel proof of the infinitude of primes using Schur's Theorem and Fermat's Last Theorem, simplifying earlier Ramsey Theory-based methods.

Findings

Proves the infinitude of primes using Schur's Theorem.

Applies the method to show other domains have infinitely many irreducibles.

Provides an alternative, simpler proof compared to previous Ramsey Theory approaches.

Abstract

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.