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 from the University of Isfahan who stated it first 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.,

$p_{n+1}^{1/(n+1)}<p_{n}^{1/n}{\text{ for all }}n\geq 1.$

Equivalently: $p_{n+1}<p_{n}^{1+{\frac {1}{n}}}{\text{ for all }}n\geq 1,$ 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 4×10¹⁸.

If the conjecture were true, then the prime gap function $g_{n}=p_{n+1}-p_{n}$ would satisfy $g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4$ and, moreover, $g_{n}<(\log p_{n})^{2}-\log p_{n}-1{\text{ for all }}n>9$; 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 is inconsistent with the heuristics of Granville and Pintz and of Maier which suggest that $g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}$ infinitely often for any $\varepsilon >0,$ where $\gamma$ denotes the Euler–Mascheroni constant.

Two related conjectures (see the comments of OEIS: A182514) are

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

which is weaker, and

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

which is stronger.

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 Edition. Springer-Verlag. ISBN 0-387-20169-6. {{cite book}}: Invalid |ref=harv (help)
• Riesel, Hans (1985). Prime Numbers and Computer Methods for Factorization, Second Edition. Birkhauser. ISBN 3-7643-3291-3. {{cite book}}: Invalid |ref=harv (help)