Firoozbakht's conjecture

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 in 1982.

The conjecture states that ${\displaystyle p_{n}^{1/n}}$ (where ${\displaystyle p_{n}}$ is the nth prime) is a strictly decreasing function of n, i.e.,

${\displaystyle {\sqrt[{n+1}]{p_{n+1}}}<{\sqrt[{n}]{p_{n}}}\qquad {\text{ for all }}n\geq 1.}$

Equivalently:

${\displaystyle p_{n+1}<p_{n}^{1+{\frac {1}{n}}},}$

${\displaystyle p_{n+1}^{n}<p_{n}^{n+1},{\text{ or }}}$

${\displaystyle \left({\frac {p_{n+1}}{p_{n}}}\right)^{n}<p_{n}.}$

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][5]

If the conjecture were true, then the prime gap function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ would satisfy:^[6]

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}\qquad {\text{ for all }}n>4.}$

Moreover:^[7]

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}-1\qquad {\text{ for all }}n>9,}$

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^[8][9][10] and of Maier^[11][12] which suggest that

${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}\approx 1.1229(\log p_{n})^{2},}$

occurs infinitely often for any ${\displaystyle \varepsilon >0,}$ where ${\displaystyle \gamma }$ 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

${\displaystyle \left({\frac {\log p_{n+1}}{\log p_{n}}}\right)^{n}<\left(1+{\frac {1}{n}}\right)^{n},}$

where the right hand side is the well-known expression which reaches Euler's number in the limit ${\displaystyle n\to \infty }$, suggesting the slightly weaker conjecture

${\displaystyle \left({\frac {\log p_{n+1}}{\log p_{n}}}\right)^{n}<e.}$

Nicholson and Farhadian^[13][14] give two stronger versions of Firoozbakht's conjecture which can be summarized as:

${\displaystyle \left({\frac {p_{n+1}}{p_{n}}}\right)^{n}<{\frac {p_{n}\log n}{\log p_{n}}}<n\log n<p_{n}\qquad {\text{ for all }}n>5,}$

where the right-hand inequality is Firoozbakht's, the middle is Nicholson's (since ${\displaystyle n\log n<p_{n}}$; see Prime number theorem § Non-asymptotic bounds on the prime-counting function), and the left-hand inequality is Farhadian's (since ${\displaystyle {\frac {p_{n}}{\log p_{n}}}<n}$; see Prime-counting function § Inequalities).

All have been verified to 2⁶⁴.^[5]