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, who stated it in 1982.
The conjecture states that (where is the -th prime) is a strictly decreasing function of ; i.e.,
for all . Equivalently, . See , .
By using a table of maximal gaps, Firoozbakht verified her conjecture up to . Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below .
If the conjecture were true, then the prime gap function would satisfy.
for all , and.
for all . 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.
Three related conjectures (see the comments of ) are variants of Firoozbakht's. Forgues notes that Firoozbakht's can be written
where the right-hand side is the well-known expression which reaches Euler's number in the limit , suggesting the slightly weaker conjecture
Nicholson and Farhadian give two stronger versions of Firoozbakht's conjecture, which can be summarized as: where is Farhadian's conjecture, is Nicholson's, is Rosser's theorem,Rosser, J. B. "The -th Prime is Greater than ". Proceedings of the London Mathematical Society 45:21-44, 1939. and is Firoozbakht's conjecture. All have been verified to 264.
See also Prime number theorem Andrica's conjecture Legendre's conjecture Oppermann's conjecture Cramér's conjecture Prime-counting function § inequalities
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