:-) Here's something else, almost entirely unrelated, that I like: we can prove that any two Fermat numbers 2^(2^n)+1 have gcd 1; this gives a proof (different

:-) Here's something else, almost entirely unrelated, that I like: we can prove that any two Fermat numbers 2^(2^n)+1 have gcd 1; this gives a proof (different from Euclid's) that there are infinitely many prime numbers: https://en.wikipedia.org/w/index.php?title=Fermat_number&oldid=1116879652#Basic_properties https://en.wikipedia.org/w/index.php?title=Fermat_number&old...