How can I show that there are infinitely many prime numbers such that $p \equiv 3 \pmod4$?
There's a hint that I'm not sure how to use, they say that lets assume that $p_1,\ldots,p_r$ are ALL the prime numbers that $ p \equiv 3 \pmod4$, look at the number:
$M = p_1p_2\cdots p_r +2$ if $r$ is even
$M = (p_1)^2p_2\cdots p_r +2$ if $r$ is odd
Prove that $M$ is a prime number that solves the equation $M \equiv 3\pmod4$.
So how can I prove that $p_1p_2\cdots p_r +2 \equiv 3 \pmod4$?