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Let $$ p_0:=2\, \ldots\, p_{n-1}\,\,p_n\,\,p_{n+1}\,\ldots $$

be the increasing sequence of all primes. How often (three questions):

  • $\,p_n > \frac{p_{n-1}+p_{n+1}}2 $
  • $\,\sqrt{p_{n-1}\cdot p_{n+1}}\, <\, p_n\, \le\, \frac{p_{n-1}+p_{n+1}}2 $
  • $\, p_n < \sqrt{p_{n-1}\cdot p_{n+1}} $

?

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1 Answer 1

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First of all, I wouldn't expect anything to be provable, but I expect you can get a good guess for the first question by using Granville's modification of Cramér's random model for prime distributions.

For the second question, assume that $p_{n+1}-p_{n-1}<\sqrt{n}/10$ (this is conjectured true for all sufficiently large $n$, but even just the prime number theorem says this is true of all but $O(\sqrt{N}\log^2 N)$ primes less than $N$).

Then $$ \frac{p_{n-1}+p_{n+1}}{2}-\sqrt{p_{n-1}p_{n+1}}<0.5 $$ so the question is asking how often $p_{n-1},p_n,p_{n+1}$ are in an arithmetic progression. The difference would have to be a multiple of 6 for all three numbers to be prime. You can get a good guess by summing over all possible multiples of six and using the Hardy-Littlewood conjecture.

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    $\begingroup$ Thank you, a fine answer, +1. (My experience tells me to still wait about accepting your answer because there is a chance that another answer may outshine yours, as unlikely as it is. Accepting could make some people disinterested.) $\endgroup$
    – Wlod AA
    Dec 23, 2019 at 18:07

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