Euclid proof for infinitely many primes congruent to seven modulo eight
= Euclid proof for infinitely many primes congruent to seven modulo eight
If $q_1,\ldots,q_r$ listed all primes congruent to seven modulo eight, then for $n=8q_1\cdots q_r$ the odd number $(n^2-2)/2$ is seven modulo eight. Every prime factor is one or seven modulo eight, at least one is seven, and none is among the $q_i$, a contradiction.