In number theory, Firoozbakht’s conjecture (or the Firoozbakht 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 (where is the nth prime) is a strictly decreasing function of n, i.e.,
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×1012. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 4×1018.
If the conjecture is true, then the prime gap function satisfies  This is one of the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than Cramer-Shanks conjecture. Moreover, Firoozbakht's conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture, but inconsistent with the heuristics of Granville and Pintz which suggest that infinitely often for any where denotes Euler's constant.
Two related conjectures (see A182514 Commments) are
which is weaker. And,
- for all values with ,
which is stronger.
Currently, Firoozbakht's conjecture is included on the List of prime conjectures involving primes numbers listed from highest to least importance. This short list of prime conjectures also includes Riemann hypothesis, Goldbach's conjecture, and Twin prime conjecture among others.
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- Gaps between consecutive primes
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