菲鲁兹巴赫特猜想

该猜想声称， pₙ^(1/n) 是一个严格递减函数（其中 pₙ 是第 n 个质数），也就是说 （公式略）

或等价地 pₙ₊₁<pₙ^(1+1/n) for all n≥ 1,

相关内容可见A182134及A246782。

借由使用最大质数间隙（maximal gap）表，法丽德·菲鲁兹巴赫特确认她的猜想对大到 4.444× 10¹² 的数都成立。利用广度更大的最大质数间隙表，目前已知该猜想对任何小于 2⁶⁴ 1.84× 10¹⁹ 的质数都成立。

若此猜想成立，那么质数间隙函数 gₙ=pₙ₊₁-pₙ 会满足下列关系： gₙ<(log pₙ)²-log pₙ for all n>4.

此外， gₙ<(log pₙ)²-log pₙ-1 for all n>9,

对此可见A111943。

该猜想是对质数间隙上界最强的猜想之一，甚至比克拉梅尔猜想和尚克斯猜想（Shanks' Conjecture）还强。从该猜想可推出强克拉梅尔猜想，而这与安德鲁·格兰维尔、平茨·亚诺什和赫尔穆特·迈尔等人的直观猜测不一致。而这些人的直观猜测认为，对任意的 ε >0 下式对无限多的数成立： （公式略）

其中 γ 是欧拉-马斯刻若尼常数。

• 两个相关的猜想（可见A182514的讨论）如下：
• 比菲鲁兹巴赫特猜想来得弱的猜想： （公式略）
• 比菲鲁兹巴赫特猜想来得强的猜想： （公式略）

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 （公式略） (where pₙ is the n -th prime) is a strictly decreasing function of n; i.e., （公式略）

for all n≥ 1. Equivalently, pₙ₊₁<pₙ^(1+1/n). See OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, 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 ≈ 1.84 × 1019.

If the conjecture were true, then the prime gap function gₙ=pₙ₊₁-pₙ would satisfy gₙ<(log pₙ)²-log pₙ

for all （公式略）, and gₙ<(log pₙ)²-log pₙ-1

for all （公式略）. See also OEIS: A111943. 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 （公式略）

occurs infinitely often for any ε >0, where γ denotes the Euler–Mascheroni constant.

Three related conjectures (see the comments of OEIS: A182514) are variants of Firoozbakht's. Forgues notes that Firoozbakht's can be written （公式略）

where the right-hand side is the well-known expression which reaches Euler's number in the limit n→ ∞, suggesting the slightly weaker conjecture （公式略）

Nicholson and Farhadian give two stronger versions of Firoozbakht's conjecture, which can be summarized as: （公式略） where A < B is Farhadian's conjecture, B < C is Nicholson's, C < D is Rosser's theorem, and A < D is Firoozbakht's conjecture. All have been verified to 264.

Applying the techniques to the known 85th maximal prime gap shows that Nicholson's (and thus Firoozbakht's) conjecture is true up to that value (1.014×1020 or 1.274×266), and any violation would require a gap greater than 2070. Likewise, a violation of Farhadian's conjecture would require a gap greater than 2066.