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.

Prime gap function

The conjecture states that ${\displaystyle {\sqrt[{n}]{p_{n}}}}$ (where ${\displaystyle p_{n}}$ is the ${\displaystyle n}$-th prime) is a strictly decreasing function of ${\displaystyle n}$; i.e.,

${\displaystyle {\sqrt[{n+1}]{p_{n+1}}}<{\sqrt[{n}]{p_{n}}}}$

for all ${\displaystyle n\geq 1}$. Equivalently, ${\displaystyle p_{n+1}<p_{n}^{1+1/n}}$. See OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, Firoozbakht verified her conjecture up to ${\displaystyle 4.444\times 10^{12}}$.^[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below ${\displaystyle 2^{64}\approx 1.84\times 10^{19}}$.^[3][4][5]

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

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}}$

for all ${\displaystyle n>4}$, and^[7]

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}-1}$

for all ${\displaystyle n>9}$. 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 the article on non-asymptotic bounds on the prime-counting function) and the left-hand inequality is Farhadian's (since ${\displaystyle p_{n}/\log p_{n}<n}$; see prime-counting function § inequalities.

All have been verified to 2⁶⁴.^[5]

See also

• Prime number theorem
• Andrica's conjecture
• Legendre's conjecture
• Oppermann's conjecture
• Cramér's conjecture