Timeline for Are polynomials bounded on the primes possible?
Current License: CC BY-SA 4.0
12 events
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| Oct 24, 2020 at 20:05 | comment | added | Yaakov Baruch | This proof trivially carries over if $P$ has real coefficients, something that does not seem to me an obvious direct consequence of the result over the rationals/integers. | |
| Oct 23, 2020 at 16:51 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 23, 2020 at 14:12 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 23, 2020 at 2:48 | comment | added | Terry Tao | One can make this argument unconditional by noting that $O(\log^{n-1-o(1)} X)$ of the $O(\log^{n-1} X)$ admissible tuples $a = (0,a_1,\dots,a_{n-1})$ with $a_1,\dots,a_{n-1}=O(\log X)$ will be associated to consecutive primes $p,p+a_1,\dots,p+a_{n-1}$ for some $p \sim X$ (because $\sim X/\log X$ primes will generate a tuple by Markov's inequality and each tuple is associated to $O(X/\log^{n-o(1)} X)$ primes by e.g. Selberg sieve). On the other hand, a polynomial constraint on these tuples would instead force at most $O(\log^{n-2} X)$ of these tuples to be admissible (Schwartz-Zippel lemma). | |
| Oct 23, 2020 at 2:40 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 23, 2020 at 1:51 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 22, 2020 at 18:08 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 22, 2020 at 15:29 | comment | added | Will Sawin | @PierrePC It's easy to amend the proof if you know a positive proportion of admissible tuples occur in the sense that all the members are prime and no numbers between them are prime. The question is entirely to do with the weaker sense of "occur" where we only demand that the members of the tuple are prime. | |
| Oct 22, 2020 at 15:23 | comment | added | Pierre PC | As Will Sawin says in another comment, Maynard's theorem implies, unconditionally, the existence of one (actually infinitely many, in a strong sense) $n$-tuple occurring infinitely often in a sequence of primes. So provided the proof can be amended, it will be unconditional. | |
| Oct 22, 2020 at 15:08 | comment | added | Yaakov Baruch | You cut right through my fog. Such simple and elegant argument! | |
| Oct 22, 2020 at 15:07 | vote | accept | Yaakov Baruch | ||
| Oct 22, 2020 at 14:58 | history | answered | Tony Huynh | CC BY-SA 4.0 |