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One of the strange proofs (among the other beautiful proof) in the book "Proofs from the book" is the fifth one, which uses a special topology on the set of integer numbers, to prove there are infinite prime numbers.

My question is:

Is this method special just for this case, or is there anything deeper and this technique (or its generalization) can be used for some other kinds of problems?

A related question (by @ToddTrimble comment):

Is Fürstenberg's topology useful?

Shahrooz
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    When unwound straightforwardly, the "topological" proof is really just the usual proof in disguise (and it doesn't use anything more than the language of topology in the first place). That said, I am under the impression that the topology introduced there is actually interesting, just not really for that reason. – Noah Schweber Mar 12 '20 at 15:27
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    If anybody could mention which topology it is, it would help answering the question. (If it's the profinite one, it definitely has a huge number of uses, say $p$-adic numbers and so on.) – YCor Mar 12 '20 at 15:55
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    Certainly it's the profinite topology, i.e., the topology inherited from the profinite completion $\hat{\mathbb{Z}}$ that appears in the adeles, as pointed out by Chandan Singh Dalawat here: https://mathoverflow.net/q/42589/2926 – Todd Trimble Mar 12 '20 at 16:00
  • @Todd Trimble It seems that I must delete this question, yes? – Shahrooz Mar 12 '20 at 16:09
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    It's your call. I'm not suggesting you should. Maybe someone would like to say something even more enlightening than the hints given so far in comments; dunno. – Todd Trimble Mar 12 '20 at 16:12
  • It is generous! – Shahrooz Mar 12 '20 at 16:14
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    PS the "Furstenberg topology" is the terminology given by some subcommunity to the profinite topology. – YCor Mar 12 '20 at 16:59
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    There is a modfication of the Furstenberg topology, called the Golomb topology (see e.g https://mathoverflow.net/q/285557). It is generated by the base consisting of the arithmetic progressions $\mathbb N\cap(a+b\mathbb Z)$ with coprime $a,b$. By the famous Dirichlet Theorem, the set of prime numbers is dense in the Golomb topology (but not in the Furstenberg one). Golomb popularized this topology expecting to find a topological proof of the famous Dirichet Theorem on density of primes. But till now such a topological proof has not been found. – Taras Banakh Mar 12 '20 at 21:10
  • @Taras I am aware of Dirichlet's Theorem on primes in arithmetic progressions, but that's not a density result. What density result do you have in mind, please? – Gerry Myerson Mar 12 '20 at 23:45
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    @GerryMyerson I had in mind the density in topological sense (as the density of the set of primes in the Golomb topology). – Taras Banakh Mar 13 '20 at 06:32

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