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Let $p_n$ be the n-th prime. The Firoozbakht Conjecture is a lesser known conjecture in the theory of primes but it has important consequences. It states that

$$ p_n^{\frac{1}{n}} > p_{n+1}^{\frac{1}{n+1}} $$

This truth of this immediately imply the Cramer's conjecture. In fact Firoozbakht conjecture is slightly better than the Cramer's conjecture in the sense that it would imply that

$$p_{n+1} - p_n < \ln^2p_n - \ln p_n.$$

Notice that while Firoozbakht Conjecture will automatically imply the Cramer conjecture, it will also disprove the Cramer-Granville Conjecture.

What has been the progress in this conjecture? Using computer calculation the conjecture has been verified, for all n upto 1.69x$10^{16}.$

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    $\begingroup$ I suppose any progress made on a conjecture of such importance would be easily located by google? $\endgroup$
    – John Jiang
    Commented Mar 6, 2012 at 4:29
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    $\begingroup$ Why would it disprove the Cramer-Granville conjecture? Gerhard "Ask Me About System Design" Paseman, 2012.03.05 $\endgroup$
    – Gerhard Paseman
    Commented Mar 6, 2012 at 5:31
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    $\begingroup$ Given the pointless discussion user 'humble' continues posting more and more comments as new answers, I vote to close as "no longer relevant". $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 15, 2012 at 18:10
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    $\begingroup$ I believe the proper reaction is to flag humble’s “answers” as spam, which I just did. $\endgroup$
    – Emil Jeřábek
    Commented Mar 15, 2012 at 18:15
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    $\begingroup$ Meta thread: tea.mathoverflow.net/discussion/1326/… $\endgroup$
    – Yemon Choi
    Commented Mar 15, 2012 at 18:33

1 Answer 1

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Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, there are investigations base on quite natural random models of the primes that contradict it. What is commonly known as Cramér's conjecture, that is that maximal gaps between consecutive primes are of sizes at most $(\log p_n)^2$ (up to lower order terms) does not contradict this conjecture, and one might even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more importantly, Granville note that a finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms). And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjecture

For details on Granville's arguments see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

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