I know this question has been asked, but I think I finally have the right proof after looking at the others. I am just confused with one part of the proof. I am confused on the part where "Any two numbers of the form 4n+1 form a product of the same form". Why would that be a contradiction?
2026-04-12 06:40:23.1775976023
Prove that there are infinitely many primes congruent to 3 modulo 4
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Your proof shows that there are infinitely many primes. Not that there are infinitely primes congruent to $3$ mod $4$. Nowhere in your proof do you mention anything about primes congruent to $3$ mod $4$, until the conclusion in the very last sentence, and this comes out of nowhere.
Instead you should start by with any finite list of primes that are congruent to $3$ mod $4$, and show that it is incomplete. That is to say, show that there must be another prime that is congruent to $3$ mod $4$, that is not in your finite list of primes congruent to $3$ mod $4$.
I will not go into the details of such a proof; these have been given many times before elsewhere on this site.