In 1982, Farideh Firoozbakht made an interesting conjecture that the sequence decreases i.e. for all ,
I present a stronger form of Firoozbakht’s conjecture.
Conjecture: Let be the n-th prime and let .
If the above conjecture is true than we can have explicit bounds on in terms of .
Corollary 1: For every , there exists a sufficiently large natural number , which depends only on , such that for all ,
The above inequality would imply Cramer’s conjecture and in fact we have a stronger bound on the gap between consecutive primes.
Corollary 2: For all sufficient large ,