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  and, moreover, ; see also A111943. This is one of the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz which suggest that infinitely often for any where denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of A182514) are
which is weaker, and
- for all values with ,
which is stronger.
- Prime number theorem
- Andrica's conjecture
- Legendre's conjecture
- Oppermann's conjecture
- Cramér's conjecture
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- Gaps between consecutive primes
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- Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture".
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- Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399.
- Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471