Fermat's Last Theorem Implies Euclid's Infinitude of Primes

TL;DR

This paper demonstrates that Fermat's Last Theorem, combined with combinatorial theorems like Schur's, implies the infinitude of primes, offering new proofs and insights linking various mathematical results.

Contribution

It shows that Fermat's Last Theorem and combinatorial theorems imply the infinitude of primes, connecting disparate areas of mathematics in a novel way.

Findings

Fermat's Last Theorem implies infinitely many primes for small exponents.

Schur's theorem and Ramsey theory contribute to Euclid's theorem.

Euclid's theorem is necessary for several key mathematical results.

Abstract

We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of Euclid's theorem, as in this case Fermat's last theorem has a proof that does not use the infinitude of primes. Similarly, we discuss implications of Roth's theorem on arithmetic progressions, Hindman's theorem, and infinite Ramsey theory towards Euclid's theorem. As a consequence we see that Euclid's Theorem is a necessary condition for many interesting (seemingly unrelated) results in mathematics.