Let $p\_n$ be the n-th prime. The [Firoozbakht Conjecture](http://www.primepuzzles.net/conjectures/conj_030.htm) 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](http://mathworld.wolfram.com/Cramer-GranvilleConjecture.html). 


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



Any progress on the Firoozbakht Conjecture?
*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](https://mathoverflow.net/questions/114399/consequences-of-legendres-conjecture/122939#122939)


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


