TITLE: Infinite quantity of primes of the form $4k+1$
QUESTION [7 upvotes]: I need to prove that there are infinitely many primes of the form $4k+1$.
I have proved that $-1$ is not a quadratic residue modulo $4k-1$ and is a quadratic residue modulo $4k+1$. Thus I need to prove that there are infinitely many primes of the form $b^2+1,\ b\in\mathbb{N}$. The standart tehnique "multiply all of them and add 1" doesn't work here. Can anybody help?

REPLY [5 votes]: Hint: What you have proven is that if $p\neq 2$, and $p|(b^2+1)$, then $p=4k+1$, since $b^2\equiv -1 \pmod{p}$.  Suppose there were finitely many primes of the form $4k+1$, say $p_1,\cdots, p_k$.  Then consider $$\left(2\prod_{i=1}^k p_k\right)^2+1.$$ What can we say about its prime divisors?  How does that imply there are infinitely many primes of the form $4k+1$?