Timeline for Primes isolated by large gaps to either side
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2015 at 9:38 | vote | accept | Joseph O'Rourke | ||
| Apr 19, 2015 at 16:20 | answer | added | GH from MO | timeline score: 5 | |
| Apr 19, 2015 at 13:19 | comment | added | Joseph O'Rourke | I would also be interested in what is currently known (in contrast to what is likely true). | |
| Apr 19, 2015 at 12:03 | comment | added | Jeremy Rouse | @StefanKohl - Yes, assuming adjacent gaps are independent. For example, the ``probability'' that the gap after $p$ has size at least $(1/2) \log^{2}(p)$ is $\int_{(1/2) \log(p)}^{\infty} e^{-x} \, dx = p^{-1/2}$. Hence the probability of two adjacent gaps of this size is about $1/p$ and $\sum_{p} \frac{1}{p}$ diverges. | |
| Apr 19, 2015 at 11:58 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Added k=1/2 data.
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| Apr 19, 2015 at 11:20 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Corrected the k=1 data.
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| Apr 19, 2015 at 10:55 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 104 characters in body
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| Apr 19, 2015 at 9:24 | comment | added | Stefan Kohl♦ | @JeremyRouse: Also for two subsequent gaps (which is what the question asks for)? | |
| Apr 19, 2015 at 2:52 | comment | added | Jeremy Rouse | The Cramer probabalistic model for gaps between primes (as described by Sound in this paper) suggests that the answer is yes if and only if $k < 2$. This is, however, way beyond what anyone can prove at this point. | |
| Apr 19, 2015 at 2:21 | comment | added | Gerhard Paseman | After some minimal checking, I got some of the bases wrong. As far as I know, not even logp_n(loglog p_n)^k is known to occur for infinitely many n and $k > 1$. I will update when I get the numbers straightened out. Maier, Pomerance, Pintz, Tao, Green, Ford, Kolyvagin, and Maynard are still some of the names to check. Gerhard "Or Use A Phone Book" Paseman, 2015.04.18 | |
| Apr 19, 2015 at 2:11 | comment | added | Gerhard Paseman | You have results of Maier and Pomerance which say there are (on average maybe?) infinitely many for some real values of $k$ larger than 1. My current investigations and various conjectures suggest your question has the answer yes only for $k$ less than 2. As a start, try Helmut Maier's Chains of large gaps between consecutive primes, done in 1981. Terry Tao announced joint work with four other authors on large gaps, available on ArXiv 1412, with (I think) an upcoming improvement on Maier's result in a followup article. Gerhard "Hope I Got Bases Right" Paseman, 2015.04.18 | |
| Apr 19, 2015 at 1:56 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
deleted 58 characters in body
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| Apr 19, 2015 at 1:52 | history | undeleted | Joseph O'Rourke | ||
| Apr 19, 2015 at 1:23 | history | deleted | Joseph O'Rourke | via Vote | |
| Apr 19, 2015 at 1:22 | history | undeleted | Joseph O'Rourke | ||
| Apr 19, 2015 at 1:09 | history | deleted | Joseph O'Rourke | via Vote | |
| Apr 19, 2015 at 0:47 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |