In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture[1][2]) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it first in 1982.
The conjecture states that p n 1 / n {\displaystyle p_{n}^{1/n}} (where p n {\displaystyle p_{n}} is the nth prime) is a strictly decreasing function of n, i.e.,
Equivalently:
see OEIS: A182134, OEIS: A246782.
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×1012.[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 264 ≈ 1.84×1019.[3][4]
If the conjecture were true, then the prime gap function g n = p n + 1 − − --> p n {\displaystyle g_{n}=p_{n+1}-p_{n}} would satisfy:[5]
Moreover:[6]
see also OEIS: A111943. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures.[4] It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz[7][8][9] and of Maier[10][11] which suggest that
occurs infinitely often for any ε ε --> > 0 , {\displaystyle \varepsilon >0,} where γ γ --> {\displaystyle \gamma } denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of OEIS: A182514) are
which is weaker, and
which is stronger.
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