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

Christian Elsholtz*

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

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.

Originalspracheenglisch
Seiten (von - bis)250-257
Seitenumfang8
FachzeitschriftAmerican Mathematical Monthly
Jahrgang128
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - 2021

Schlagwörter

  • prime numbers
  • Schur's theorem
  • Fermat's Last Theorem

ASJC Scopus subject areas

  • !!Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

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