This conjecture was first stated by the mathematician
Farideh Firoozbakht from the University of Isfahan.
It appeared in print in The Little Book of Bigger Primes by Paulo Ribenboim (2004, page 185).
The Firoozbakht conjecture is one of the strongest upper bounds for prime gaps
even somewhat stronger
than the Cramér-Shanks conjecture
(predicting that the gaps near x are at most about as large as ln²x).
The precise formulation is as follows:
The Firoozbakht conjecture.
Let pk be the k-th prime, then the sequence
(pk)1/k is strictly decreasing.
(pk)k+1 > (pk+1)k,
where pk is the k-th prime.
As of 2014, a rigorous proof of the conjecture is not known nor do we have any counterexamples.
The conjecture has been verified up to 4×1018
If the conjecture is true,
pk+1 − pk
< ln²pk − ln pk − 1
for k > 9