Timeline for For what subsets S of (Z/nZ)* is there a Euclidean proof that there are infinitely many primes whose residues lie in S?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Jun 8 at 1:06 | history | protected | Yemon Choi | ||
| Jun 6 at 14:02 | comment | added | LSpice | The two different meanings of $p$ in $p(p_1... p_k)$ have been bugging me, so I changed it to $P(p_1\dotsm p_k)$ while this was back on the front page. I hope that was OK. | |
| Jun 6 at 14:00 | history | edited | LSpice | CC BY-SA 4.0 |
p -> P, and name of Conrad paper, while this is on the front page
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| Jun 6 at 12:17 | history | edited | Vincent | CC BY-SA 4.0 |
updated link
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| Jan 2, 2021 at 0:39 | comment | added | Elaqqad | This was asked ten years ago, but it is one of my concerns right now. But more broadly, I am not interested only in the case where $p$ is a polynomial, it can be just defined algebraically or combinatorially. I still don't understand why do we need to push in analysis in the problem (also I agree that analytic number theory proved to be very fruitful, but It feels to me that if we use algebraic methods they should be more generalizable) | |
| Jan 14, 2011 at 6:20 | comment | added | Pete L. Clark | I feel that Frictionless Jellyfish's comment is a little too harsh. Of course the analytic proof of Dirichlet's theorem is important and deserving of coverage in courses (in fact, it is up there with the most important proofs in the history of number theory, and my esteem for Dirichlet for being able to come up with this argument in the 1830's (!!!) is boundless). But searching for elementary proofs and trying to understand the range of applicability of the Euclidean argument are also very interesting: at least they are to me. Why not ask about them? I hardly think it's a waste of time. | |
| Jan 13, 2011 at 18:16 | comment | added | Joe Silverman | There was a great deal of excitement at the time when Erdos and Selberg came up with an elementary (i.e., not using complex analysis) proof of the prime number theorem. The hope, of course, was that the new proof would provide new insights, and maybe even give better error estimates. Similarly, I think there would be significant interest in an elementary proof of the general form of Dirichlet's theorem. Of course, elementary does not mean easy, or suitable for teaching to high school students. I agree that for teaching, the analytic proofs are best. | |
| Jan 13, 2011 at 17:15 | answer | added | David E Speyer | timeline score: 13 | |
| Jan 13, 2011 at 14:53 | answer | added | Asterios Gkantzounis | timeline score: 1 | |
| Dec 25, 2010 at 15:08 | comment | added | KConrad | Qiaochu: what you're gaining from analysis is the full theorem. I wouldn't consider the high school methods as doing anything for you but giving some special cases, which do provide a nice illustration of some special cases of quadratic reciprocity (e.g., knowing when -1 mod p is a square or when 2 mod p is a square can be used to prove special cases of Dirichlet's theorem mod 4 or mod 8) but not much of value beyond that I think. | |
| Dec 25, 2010 at 6:19 | comment | added | Qiaochu Yuan | There are a couple of reasons. The first is pedagogical: it's nice to know how much of Dirichlet's theorem you can prove to a group of bright high school students without (perhaps I should say before) introducing analysis. The second is that I am interested in the "limit" of elementary methods so I can better appreciate exactly what I am gaining from using analysis. | |
| Dec 24, 2010 at 18:33 | comment | added | Terry Tao | This does not directly answer your question, but in Granville's paper ams.org/mathscinet-getitem?mr=1220462 elementary methods are used to show that the primes mod n are equidistributed either in all of $(Z/NZ)^*$ or in the complement of an index 2 multiplicative subgroup containing the squares, and that this is in some sense the "limit" of elementary methods. This suggests (but does not prove) that for your question, S cannot be contained in any index 2 multiplicative subgroup of $(Z/NZ)^*$. | |
| Dec 24, 2010 at 16:07 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
deleted 21 characters in body; added 44 characters in body
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| Dec 24, 2010 at 16:02 | comment | added | Qiaochu Yuan | Sure. I messed up my example and had to rethink the question a bit, but I think everything works now. | |
| Dec 24, 2010 at 16:01 | history | undeleted | Qiaochu Yuan | ||
| Dec 24, 2010 at 16:01 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 944 characters in body; deleted 53 characters in body
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| Dec 24, 2010 at 15:31 | history | deleted | Qiaochu Yuan | ||
| Dec 24, 2010 at 15:26 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 947 characters in body; deleted 1140 characters in body
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| Dec 24, 2010 at 15:06 | comment | added | KConrad | Please give an example or two of such subsets S, to improve the context of the question. | |
| Dec 24, 2010 at 15:04 | history | edited | KConrad | CC BY-SA 2.5 |
added 21 characters in body
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| Dec 24, 2010 at 14:39 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |