= Euclid proof for infinitely many primes congruent to seven modulo eight
{c}
Suppose that $q_1,\ldots,q_r$ listed all <prime numbers> congruent to seven modulo eight. Put $m=q_1\cdots q_r$, with the empty <product> equal to one, and define
$$
N=32m^2-1\equiv7\pmod8.
$$
If an <odd prime> $\ell$ divides $N$, then $(8m)^2\equiv2\pmod\ell$, so the <second supplementary law for quadratic reciprocity> forces $\ell\equiv1$ or $7\pmod8$. Since $N$ is seven modulo eight, at least one of its <prime factors> is seven modulo eight. But $N\equiv-1\pmod{q_i}$ for every listed <prime>, so this <prime factor> is new. The contradiction proves infinitude.
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