Show that there are infinitely many primes of the form $8n+1,8n+3,8n+5,8n+7$

Say you have $p|x^2+2$ implies $p\in\{1,3\}\bmod 8$. Now suppose there are only finitely many primes of the form $8n+3$. Let $\Pi$ be the product of these primes and consider the combination

$M=\Pi^2+2$

None of the primes used to make $\Pi$ can be a factor of $M$ and the actual prime factors of $M$ cannot be all of the form $8n+1$ because $M\equiv 3\bmod 4$. We are forced to allow more $8n+3$ prime factors, thus the proposed finite set of such primes could not have contained all of them.

Use your other expressions to render similar proofs for the other cases.