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$10^{16}.$