I am almost done with my proof that there are infinitely many primes of the form $4k+3$, but have one remaining concern.
My chosen $N$, the subject of my contradictions, is defined as $4(p_1p_2\cdots p_n-1)+3$ where $p_1p_2\cdots p_n$ are the finite primes of the form $4k+3$ (assumed finite earlier for contradiction.)
My proof hinges on the fact that none of the $p_i$'s can divide $N$ without remainder. I claim that this is by "construction of $N$," but lack the more rigorous proof. Any tips?