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.
- MSC: Primary 11A41
- Secondary 05D10
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- Information, Communication & Computing