Firoozbakht's conjecture
In number theory, Firoozbakht’s conjecture is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht from the University of Isfahan who stated it first in 1982.
The conjecture states that is a strictly decreasing function of n, i.e.,
Equivalently:
see,.
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 264 ≈.
If the conjecture were true, then the prime gap function would satisfy:
Moreover:
see also. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz and of Maier which suggest that
occurs infinitely often for any where denotes the Euler–Mascheroni constant.
Two related conjectures are
which is weaker, and
which is stronger.
