There are at least two well known proofs of the infinitude of primes (Euclid's original one and Euler's proof using L-series) and both of them can be extended to prove more general statements of the form: "There exist infinitely many primes of the form an+b" for specific $a,b$.

Here is another proof that there are infinitely many primes using algebro-geometric ideas:

Suppose there were only finitely many primes. Then $\operatorname{Spec} \mathbb Z$ would be an artin ring and in particular, any regular ring finite over it would be locally a PID and Artin, hence globally a PID.

In particular, this would imply that all (integrally closed) number rings have class number one which is clearly false, hence there would have to be infinitely many primes.

I don't believe this is a circular argument! (This idea is definitely not original to me but I sadly don't remember where I saw this argument. I very vaguely remember reading it from a post of David Speyer on this very site...)

Question: Does anyone see how to extend this to proving that there are infinitely many primes of the form $4n+1$ or $4n+3$? Or more general statements of this form.

  • $\begingroup$ The first place I checked for this was mathoverflow.net/questions/42512/… (I didn't check carefully; I was just (ctrl-F)-ing with David Speyer's name. But if it's not there, it might be a good candidate. (-: ) $\endgroup$ – Todd Trimble Nov 22 '18 at 23:25
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    $\begingroup$ Your argument goes back to Larry Washington, who pointed out that since $\mathbf Z[\sqrt{-5}]$ is not a UFD there have to be infinitely many primes. See the list of many proofs of infinitude of the primes in Ribenboim's Book of Prime Number Records or New Book or Prime Number Records. $\endgroup$ – KConrad Nov 22 '18 at 23:37
  • $\begingroup$ @KConrad Thanks for explaining the history! $\endgroup$ – Asvin Nov 22 '18 at 23:49

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