  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.
