Infinitely Many Primes of the Form $4n+3$

Statement

There exist infinitely many primes $p$ such that \[ p \equiv 3 \pmod 4. \]

Proof

Assume finitely many such primes: \[ p_1,\dots,p_k \equiv 3 \pmod 4. \] Define \[ N = 4p_1p_2\cdots p_k - 1. \] Then \[ N \equiv -1 \equiv 3 \pmod 4, \qquad N \equiv -1 \pmod{p_i}\ \Rightarrow\ p_i\nmid N. \] Let $q$ be a prime divisor of $N$. If all prime divisors of $N$ satisfied $q \equiv 1 \pmod 4$, then their product would satisfy \[ N \equiv 1 \pmod 4, \] contradiction. Hence $\exists q\mid N$ with \[ q \equiv 3 \pmod 4. \] But $q\ne p_i$ for all $i$, since $p_i\nmid N$. Contradiction. Therefore infinitely many.