State the prime number theorem, and Dirichlet's theorem on primes in arithmetic progression.

If $p$ is an odd prime number, prove that $-1$ is a quadratic residue modulo $p$ if and only if $p \equiv 1 \bmod 4$.

Let $p_{1}, \ldots, p_{m}$ be distinct prime numbers, and define

$$N_{1}=4 p_{1} \ldots p_{m}-1, \quad N_{2}=4\left(p_{1} \ldots p_{m}\right)^{2}+1$$

Prove that $N_{1}$ has at least one prime factor which is congruent to $3 \bmod 4$, and that every prime factor of $N_{2}$ must be congruent to $1 \bmod 4$.

Deduce that there are infinitely many primes which are congruent to $1 \bmod 4$, and infinitely many primes which are congruent to $3 \bmod 4$.