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Let $(p_{n})_{n∈ℕ}$ be the sequence of consecutive primes. In P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1995, page 185, the author says:

A new conjecture by F. Firoozbakht, dating from about 1982, was communicated to me by the author; as far as I know, it remains unpublished. The conjecture is that the sequence $(p_{n}^{(1/n)})_{n≥2}$ is strictly decreasing.

Have there been any new developments in this direction?

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    $\begingroup$ mathoverflow.net/questions/90327/… - I had an impression from discussing this with various people back then that this conjecture is very unlikely to be true... $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 8, 2014 at 17:51
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    $\begingroup$ I voted to leave this open because the "duplicate" question was nearly 3 years old. $\endgroup$
    – Karl Schwede
    Commented Dec 8, 2014 at 23:00
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    $\begingroup$ @KarlSchwede As explained in Carlo's answer, this is slighty stronger than a 80-year old conjecture of Cramer, which is NOT a consequence of the Riemann hypothesis. I don't think you should expect new developments every few years and, if there are, you'll hear about them. $\endgroup$
    – Felipe Voloch
    Commented Dec 9, 2014 at 14:54
  • $\begingroup$ That's true, but the other question was closed. $\endgroup$
    – Karl Schwede
    Commented Dec 10, 2014 at 16:14
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    $\begingroup$ @KarlSchwede: indeed, and that question was closed for a reason - the OP for that question certainly did not follow any conventional etiquette patterns, to put it mildly. In any case, you can see that the answer to that question was updated in 2013, so not that long ago. Plus, the very first comment to that question states, "I suppose any progress made on a conjecture of such importance would be easily located by google", and I could not agree more. $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 10, 2014 at 18:37

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The status as of 2010 is summarized in On a new property of primes that leads to a generalization of Cramer's conjecture . (Firoozbakht's conjecture is slightly stronger than Cramér's, as explained here.)

I find it noteworthy that Farideh Firoozbakht proposed this conjecture as an undergraduate student.

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