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Is there a nice elementary way to prove there is infinite prime numbers of form $5n+3$ (also for $5n+2$) with $n\in \mathbb{N}$?

I know how to do it for primes of form $pn+1$ for any prime $p\geq 3$ but not in this case.

I'm also familiar with the theorems of Schur and Murty.

Martin Sleziak
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User2020201
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    This would be a good question for math.stackexchange. – Laurent Moret-Bailly Apr 29 '20 at 07:55
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    The question https://mathoverflow.net/questions/16735/is-a-non-analytic-proof-of-dirichlets-theorem-on-primes-known-or-possible is a more general version of this (and to my opinion suitable for this forum). It refers to https://kconrad.math.uconn.edu/blurbs/gradnumthy/dirichleteuclid.pdf for the results of Schur and Murty and more. The question https://mathoverflow.net/questions/15220/is-there-an-elementary-proof-of-the-infinitude-of-completely-split-primes is also relevant. – Chris Wuthrich Apr 29 '20 at 09:53
  • Why? Is it so trivial to answer this and not worth of a research? @LaurentMoret-Bailly Anyway, I did post it on MSE. – User2020201 Apr 29 '20 at 10:16
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    To add to the previous comment, here is link to the post on [math.se]: How to prove there is infinite prime numbers of form $5n+3$ without Dirichlet theorem? – Martin Sleziak Apr 29 '20 at 12:04
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    In my opinion, this question would clearly be on-topic here, if it weren't for that Konstantinos's answer in Chris's link also answers this question. It is clearly not trivial with $5$ replaced with an arbitrary integer, evidenced by the fact that no such proof has appeared in the linked question (despite having 60 upvotes) and I personally consider the $m=5$ proof there also quite non-trivial. The question also demonstrates awareness of relevant literature, and I would say is of general interest to mathematicians (or at least to me)... but it is indeed arguably a duplicate. – dhy Apr 29 '20 at 12:37
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    I wouldn't necessarily consider this a research-level question, but it's certainly not trivial. One can certainly adapt the proof of Dirichlet's Theorem to this special case (checking $L(\chi,s)\neq 0$ should be straightforward in this specific case), but this is probably not the kind of answer OP wants. – Wojowu Apr 29 '20 at 12:40
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    To confirm what Wojowu says: For the quadratic character $\chi$, $\sum_{n=1}^{\infty} \tfrac{\chi(n)}{n} = \sum_{m=0}^{\infty} \left( \tfrac{1}{5m+1} - \tfrac{1}{5m+2} - \tfrac{1}{5m+3} + \tfrac{1}{5m+4} \right)= \sum_{m=0}^{\infty} \tfrac{10 (2m+1)}{(5m+1)(5m+2)(5m+3)(5m+4)}$ and all summands are positive. – David E Speyer Apr 29 '20 at 13:03
  • Since you are familiar with Schur and Murty's theorems, you seem to be asking for an elementary proof that doesn't use a Euclidean polynomial argument. Let me ask then slightly more generally, do you know of any elementary proof of this sort for any residue class that doesn't rely on a polynomial in that way? I don't know of any, but it would seem that trying to find one in a simple case would be an obvious starting point. – JoshuaZ Apr 29 '20 at 13:04
  • For a nonreal character $\epsilon$, we have $\sum_{n=1}^{\infty} \tfrac{\epsilon(n)}{n} = \sum_{m=0}^{\infty} \left( \tfrac{1}{5m+1} \pm \tfrac{i}{5m+2} \mp \tfrac{i}{5m+3} - \tfrac{1}{5m+4} \right) = \sum_{m=0}^{\infty} \tfrac{3}{(5m+1)(5m+4)} \pm i \sum_{m=0}^{\infty} \tfrac{1}{(5m+2)(5m+3)}$, and the summands are similarly positive. – David E Speyer Apr 29 '20 at 13:05
  • No, I also don't know any such proof and would like to see it if exists. @JoshuaZ I was thinking to present my hischool students this Euclidian type of proofs and then I wonder if there are also some others types. Now it seem that I should dive into the proof of Dirichlet theorem itself in these special cases. – User2020201 Apr 29 '20 at 13:09
  • Other question threads on this topic have been linked to above. In one of those, I mention in a comment that there are proofs of Bang, Ricci, Roux, and Erdos that prove Dirichlet's theorem for certain special progressions by methods analogous to those used by Chebyshev to study the distribution of primes. Erdos's argument, for instance, is certainly "elementary" in the technical sense but ... I don't think any of the mentioned proofs are "simple". For your purposes it might be better to just specialize Dirichlet's proof in the ways suggested above. – so-called friend Don Apr 29 '20 at 17:49
  • I incorrectly believed that Euclid's argument would work in this case, but it doesn't. I don't know of a proof that's materially different than Dirichlet's proof. – Stanley Yao Xiao Apr 29 '20 at 19:09
  • Ajjaja @LaurentMoret-Bailly Is that only contribution to this question, to vote for close down? I'm very disappointed for such a manner. – User2020201 Apr 30 '20 at 13:30

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