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

Christian Elsholtz

doi:10.1080/00029890.2021.1856544

Abstract

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 toward Euclid’s theorem. As a consequence we see that Euclid’s theorem is a necessary condition for many interesting (seemingly unrelated) results in mathematics.

Highlights

Imagine that the set of positive integers has only finitely many primes
If you are a number theorist, you will realize that a major part of analytic number theory just vanishes
One of the implications of this article is that algebraic number theorists and combinatorialists would live in a very different world, too

Summary

INTRODUCTION

Imagine that the set of positive integers has only finitely many primes. We will investigate consequences, and to become more creative with this, we imagine we live in an entirely different world, namely in a “world with only finitely many primes.” If you are a number theorist, you will realize that a major part of analytic number theory just vanishes. There are many proofs of Euclid’s theorem stating that there exist infinitely many primes. For an application of Schur’s theorem it is perfectly fine if an integer m with hypothetical distinct prime factorizations is assigned only one of the colors. We prove this by dividing a lower bound approximation of the number of nth powers in [1, N ] by an upper bound approximation of all integers in [1, N ], both counted by means of exponent patterns. Log N nk log pk of all integers at most N are nth powers, which gives (for large N ) a positive proportion of at least δ ≥ With this lemma we can replace Schur’s theorem by Roth’s theorem. We note that the main focus of this article is not about short proofs but how seemingly remote results can be applied

DISCUSSION

CONCLUSION