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In this post for integers $n\geq 1$ we denote the $n$-th prime number as $p_n$.

When we consider that $k>1$ runs over integers, from the theory of the Stolarsky mean we can deduce that as $k\to \infty$ the inequality $$\frac{n(p_{n+1}-p_n)}{\log p_n}<\left(\frac{k\left(\sqrt[k]{p_{n+1}}-\sqrt[k]{p_{n}}\,\right)}{p_{n+1}-p_n}\right)^{-k/(k-1)}\tag{1}$$

becomes in the known as Firoozbakht's conjecture, because as $k\to \infty$ the RHS of previous inequality tends to the logarithmic mean of consecutive primes numbers $p_{n+1}$ and $p_{n}$.

Wikipedia has an article for Stolarsky mean in which I was inspired (that is the specialization for $p=1/k$ from the quotient that defines the Stolarsky mean and after I do a comparison with the logarithmic mean of the section Special cases). For the mentioned conjecture ([1]) Wikipedia has the article Firoozbakht's conjecture that refers in the first paragraph the history of this conjecture.

Question. I would like to know what is (approximately) the largest integer $k>1$ for which we can to prove that $(1)$ is true $\forall n\geq 1$ or $\forall n\geq n_0$ with $n_0$ a suitable choice ($n_0=n_0(k)$ maybe depending on $k$) from your calculations. Many thanks.

I know how grows the logarithmic mean of consecutive prime numbers by the squeeze theorem (I mean the inequality between the logarithmic mean, and the geometric and arithmetic means) and invoking the prime number theorem, but I believe that it will not be enough to approach our Question, and I know that the questions about prime gaps $g_n:=p_{n+1}-p_n$ are very difficult.

References:

[1] Conjecture 30 The Firoozbakht Conjecture, Carlos Rivera's The Prime Puzzles & Problems Connection, (2012).

[2] Kenneth B. Stolarsky, Generalizations of the logarithmic mean, Mathematics Magazine. 48, (1975), 87–92.

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    $\begingroup$ The prime gaps have been studied upto a very large search limit. For larger numbers, we only have weak upper bounds for the prime gap, probably not enough to approve $(1)$ $\endgroup$ – Peter Apr 20 at 7:46
  • $\begingroup$ Many thanks for your excellent contribution. Feel free to add an answer if one can to provide an argument about my Question from the known tables for prime gaps @Peter . Also this comment is an invitation if you want to visit my other posts in this site Mathematics Stack Exchange: it is a great help for me if you comment from your knowledges or computations if those posts are interesting. $\endgroup$ – user759001 Apr 20 at 7:49
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    $\begingroup$ If you can derive bounds for the so-called merit of a prime gap, this has been extensively calculated. Most records are hold by Dana Jacobsen. $\endgroup$ – Peter Apr 20 at 7:51
  • $\begingroup$ I didn't know what is the notion of merit for prime gaps. I'm going to read the definition @Peter $\endgroup$ – user759001 Apr 20 at 7:52
  • $\begingroup$ Also one can to interpret factors in both sides of Firoozbakht's conjecture as geometric means: for example the factor $p_n^{1/n}$ can be interpreted as $$\sqrt[n]{\alpha^{n-1}(1+\beta)^{n-1}p_n}$$ (for the specialization of the functions $\alpha=\alpha(n)=1$ and $\beta=\beta(n)=0$ $\forall n\geq 1$) that is thus the geometric mean of the $n-1$ factors $\alpha(1+\beta)$ multiplied by the factor $p_n$. This isn't related to my post, are addtional attempts to get other weaker forms of the Firoozbakht's conjecture for some $\alpha(n)>1$ and/or $\beta(n)>0$ @Peter Thanks again. $\endgroup$ – user759001 Apr 28 at 20:20

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