Primes of the form $4n+1$

Setup. Suppose finitely many primes $p_i\equiv1\pmod4$. Let $N=(p_1\cdots p_n)^2+1$.

Mod 4. $N\equiv1+1\equiv2\pmod4$. Any odd prime divisor $q$ of $N$ has $p_1^2\cdots p_n^2\equiv-1\pmod q$, so $-1$ is a QR mod $q$. By Euler's criterion, this forces $q\equiv1\pmod4$.

Contradiction. But $q$ must be one of the $p_i$, and $p_i\mid N$ would give $p_i\mid1$, contradiction.

Conclusion. Infinitely many primes $\equiv1\pmod4$.