Awhile ago in class, we were shown a nifty proof that there were infinite primes of the form $4k+1$.

But then for $4k+3$, we used a theorem that was never proven to us (Dirichlet's Theorem).

So here is my own attempt to prove the same statement. Tell me what you think.

Assume there are finitely ($n$) many primes of the form $4k+3$: $p_{1}, \dots, p_{n}$

Let $N = 4(p_{1}, \dots, p_{n}) - 1$

This odd number can be factored into prime divisors.

But, it cannot be that all prime factors have the form $4k+1$ because as we showed in class, the product of numbers of the form $4k+1$ still has the form $4k+1$.

Therefore, there must be a prime divisor of the form $4k+3$. Thus, $N$ is divisible by one of $p_{1}, \dots, p_{n}$ since these are all the primes in the form $4k+3$.

But this implies that $p_{i} \mid 1$ for $1 \leq i \leq n$.

And this is impossible. $\to \gets$