Timeline for Topology and infinite number of primes
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 13, 2020 at 6:32 | comment | added | Taras Banakh | @GerryMyerson I had in mind the density in topological sense (as the density of the set of primes in the Golomb topology). | |
| Mar 12, 2020 at 23:45 | comment | added | Gerry Myerson | @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? | |
| Mar 12, 2020 at 21:10 | comment | added | Taras Banakh | There is a modfication of the Furstenberg topology, called the Golomb topology (see e.g 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. | |
| Mar 12, 2020 at 17:15 | review | Close votes | |||
| Mar 17, 2020 at 3:05 | |||||
| Mar 12, 2020 at 16:59 | comment | added | YCor | PS the "Furstenberg topology" is the terminology given by some subcommunity to the profinite topology. | |
| Mar 12, 2020 at 16:35 | history | edited | Shahrooz | CC BY-SA 4.0 |
added 125 characters in body
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| Mar 12, 2020 at 16:14 | comment | added | Shahrooz | It is generous! | |
| Mar 12, 2020 at 16:12 | comment | added | Todd Trimble | 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. | |
| Mar 12, 2020 at 16:09 | comment | added | Shahrooz | @Todd Trimble It seems that I must delete this question, yes? | |
| Mar 12, 2020 at 16:00 | comment | added | Todd Trimble | 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: mathoverflow.net/q/42589/2926 | |
| Mar 12, 2020 at 15:55 | comment | added | YCor | 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.) | |
| Mar 12, 2020 at 15:54 | history | edited | YCor | CC BY-SA 4.0 |
fixed English
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| Mar 12, 2020 at 15:27 | comment | added | Noah Schweber | 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. | |
| Mar 12, 2020 at 15:09 | history | asked | Shahrooz | CC BY-SA 4.0 |