I was able to prove that there are infinitely many prime numbers but I'm not able prove that there are infinitely many primes that are equivalent to $3 \pmod{4}$. How do I go about doing this?
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If there are finitely many primes $q_i \equiv 3 \bmod 4$ look at $4 \prod_{i=1}^4 q_i-1$. It works because $\varphi(\varphi(n)) = 1$. – reuns Oct 16 '17 at 03:51