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.
Originalsprache | englisch |
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Seiten (von - bis) | 250-257 |
Seitenumfang | 8 |
Fachzeitschrift | American Mathematical Monthly |
Jahrgang | 128 |
Ausgabenummer | 3 |
DOIs | |
Publikationsstatus | Veröffentlicht - 2021 |
Schlagwörter
- prime numbers
- Schur's theorem
- Fermat's Last Theorem
ASJC Scopus subject areas
- Mathematik (insg.)
Fields of Expertise
- Information, Communication & Computing