Bound on the gaps between prime numbers
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 formula_1 (where formula_2 is the "n"th prime) is a strictly decreasing function of "n", i.e.,
formula_3
Equivalently:
formula_4
formula_5
formula_6
see OEIS: , OEIS: .
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 264 ≈ .
If the conjecture were true, then the prime gap function formula_7 would satisfy:
formula_8
Moreover:
formula_9
see also OEIS: . 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
formula_10
occurs infinitely often for any formula_11 where formula_12 denotes the Euler–Mascheroni constant.
Three related conjectures (see the comments of OEIS: ) are variants of Firoozbakht's. Forgues notes that Firoozbakht's can be written
formula_13
where the right hand side is the well-known expression which reaches Euler's number in the limit formula_14, suggesting the slightly weaker conjecture
formula_15
Nicholson and Farhadian give two stronger versions of Firoozbakht's conjecture which can be summarized as:
formula_16
where the right-hand inequality is Firoozbakht's, the middle is Nicholson's (since formula_17; see ), and the left-hand inequality is Farhadian's (since formula_18; see ).
All have been verified to 264.
Notes.
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