Let $p_n$ be the n-th prime. The [Firoozbakht Conjecture][1] 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][2]. 

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


  [1]: http://www.primepuzzles.net/conjectures/conj_030.htm
  [2]: http://mathworld.wolfram.com/Cramer-GranvilleConjecture.html