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Say that the $n$-th prime $p_n$ is isolated to degree $k$ (my notation) if the prime gap to either side is larger than $\log p_n$ to the $k$-th power: \begin{eqnarray*} p_n - p_{n-1} & > & (\log p_n)^k \;,\\ p_{n+1} - p_n & > & (\log p_n)^k \;, \end{eqnarray*} where $\log$ is the natural log.

Examples.

  • For $k=1.5$, $p_{4059}=38501$ is isolated because $$(p_{n-1},p_n,p_{n+1}) = (38461,38501,38543)\;,$$ and the gaps of $40$ and $42$ both exceed $(\log 38501)^{3/2} \approx 10.6^{1.5} \approx 34.3$.

  • For $k=1.6$, $p_{722697}=10938023$ is isolated because $$(p_{n-1},p_n,p_{n+1}) = (10937921,10938023,10938119)\;,$$ and the gaps of $102$ and $96$ both exceed $(\log 10938023)^{1.6} \approx 86.2$.

  • For $k=1.7$, I find no isolated primes in the first $10$-million primes. (The $10$-th million prime is $179424673$.)

  • Among the first $10$-million primes, about $13$% are isolated to degree $k = 1$, and $73$% are isolated to degree $k=\frac{1}{2}$.

Q. For which $k$ are there an infinite number of isolated primes of degree $k$?

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  • $\begingroup$ 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 $\endgroup$ – Gerhard Paseman Apr 19 '15 at 2:11
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    $\begingroup$ 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 $\endgroup$ – Gerhard Paseman Apr 19 '15 at 2:21
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    $\begingroup$ 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. $\endgroup$ – Jeremy Rouse Apr 19 '15 at 2:52
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    $\begingroup$ @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. $\endgroup$ – Jeremy Rouse Apr 19 '15 at 12:03
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    $\begingroup$ I would also be interested in what is currently known (in contrast to what is likely true). $\endgroup$ – Joseph O'Rourke Apr 19 '15 at 13:19
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Currently it is not even known, for any $k>1$, if $p_{n+1}-p_n>(\log p_n)^k$ holds infinitely often. The best known result in this direction is due to Ford, Green, Konyagin, Maynard, Tao, see here.

On the other hand, as Jeremy Rouse explained in two comments, it is expected that the original inequalities hold infinitely often for any $k<2$.

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  • $\begingroup$ In the document behind your link, they hint at an answer to Joseph's question. Do you know if that writeup has appeared, even in prearxiv form? $\endgroup$ – The Masked Avenger Apr 19 '15 at 16:25
  • $\begingroup$ @TheMaskedAvenger: Can you be more specific, with exact page and line number in the document arxiv.org/pdf/1412.5029v2.pdf ? $\endgroup$ – GH from MO Apr 19 '15 at 17:16
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    $\begingroup$ No, but between Theorem 1 and the section 1.1 they cite a sequel which extends the Maier matrix method. I may be misreading things, so I ask you, since you seem more able with the analytic number theory literature (compared to me, which isn't saying much). $\endgroup$ – The Masked Avenger Apr 19 '15 at 19:02
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    $\begingroup$ @TheMaskedAvenger: Thanks for the clarification. I think this writeup has not appeared, but I have no doubt it will. The way I interpret the authors' remark is as follows: given any $m>1$, there are infinitely many $n$'s so that the minimal gap size between the primes $p_n,\dots,p_{n+m}$ is significantly larger than $\log n$, perhaps as large as the bound in Theorem 1 of the paper (corresponding to $m=1$). The matrix method will play a role in the proof of this result. $\endgroup$ – GH from MO Apr 19 '15 at 19:33
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    $\begingroup$ @GerhardPaseman: Indeed, your version of the inequality is not known to hold for infinitely many $n$'s when $k=1$, but it is known to hold when $k<1$. $\endgroup$ – GH from MO Apr 19 '15 at 21:53

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