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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? enter image description here

Advent21
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  • start slowly... what prime congruent to $3 \pmod 4$ divides $4 \cdot 3 -1 ; ? ;$ Same question for $4 \cdot 3 \cdot 7 -1 ; ? ;$ – Will Jagy Oct 14 '19 at 01:16
  • once you have those, same question for $4 \cdot 3 \cdot 7 \cdot 11-1 ; , ;$ where we finally get some variation, in that the number specified is composite – Will Jagy Oct 14 '19 at 01:20
  • Cf. https://math.stackexchange.com/questions/1280254/relevance-of-prime-being-divisble-by-4k1-in-proof-that-there-are-infinitely?rq=1 – Travis Willse Oct 14 '19 at 01:40

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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.

Servaes
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  • Please don’t recommend a proof by fake contradiction. One may simply assume that you have a finite list of primes congruent to $3$ modulo $4$, and conclude that there is a prime congruent to $3$ modulo $4$ not on the list. This in analogy to the original proof that there are infinitely many primes, which just shows that any finite list is incomplete. – Arturo Magidin Oct 14 '19 at 02:23
  • I’ve used the term before. A “fake proof by contradiction” (or proof by “fake contradiction”) is a proof, either direct or by contrapositive, in which we add “suppose to the contrary that the conclusion is false” at the top, and then add “which contradicts our assumption” at the bottom. Such as the standard modern presentation of the infinitude of primes, or the standard presentation of Cantor’s Diagonal argument. The extra assumption is never used, except to contradict the conclusion... – Arturo Magidin Oct 14 '19 at 02:44
  • See here, and especially here. – Arturo Magidin Oct 14 '19 at 02:45
  • Okay, thank you. I think I finally got it, but I am confused on one part of my improved proof. Why is "Any two numbers of the form 4n+1 form a product of the same form". a contradiction? – Advent21 Oct 17 '19 at 23:29