Firoozbakht's conjecture

Prime gap function

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

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

Equivalently:

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

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

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

see OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×10¹². 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 $g_{n}=p_{n+1}-p_{n}$ would satisfy:

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

Moreover:

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

see also OEIS: A111943. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. 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

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

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

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

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

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

Nicholson and Farhadian give two stronger versions of Firoozbakht's conjecture which can be summarized as:

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

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

All have been verified to 2⁶⁴.

See also

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

References

• Ribenboim, Paulo (2004). The Little Book of Bigger Primes (Second ed.). Springer-Verlag. ISBN 0-387-20169-6.
• Riesel, Hans (1985). Prime Numbers and Computer Methods for Factorization (Second ed.). Birkhauser. ISBN 3-7643-3291-3.