In this question, I have to prove that there exists infinitely many primes for $n = 4k - 1$. We can consider this sequence $3, 5, 7, 11, 19, 23, 31, \ldots$, which are all a form of $4k - 1$.
So, we start by considering the positive integers of the form $n=4k-1$ and their possible prime divisors.
Show that every prime divisor of $4k - 1$ is odd.
Show that an odd number is either of the form $4k - 1$ or $4k + 1$.
Show that a product of two numbers of the form $4k + 1$ again has that form (more precisely, given two numbers $4k + 1$, $4l + 1$, show that their product has the form $4m + 1$ for some integer $m$).
I need help up to this point to solve the rest of the question. Any pointers will be greatly appreciated!