# Firoozbakht's conjecture

Prime gap function

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 from the University of Isfahan who stated it first in 1982.[3]

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 p_{n+1}^{1/(n+1)}

Equivalently: ${\displaystyle p_{n+1} see , .

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 4×1018.[4]

If the conjecture is true, then the prime gap function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ satisfies ${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4.}$[5] This is one of the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than Cramer-Shanks conjecture.[6] Moreover, Firoozbakht's conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture,[7] but inconsistent with the heuristics of Granville and Pintz[8][9][10] which suggest that ${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}}$ infinitely often for any ${\displaystyle \varepsilon >0,}$ where ${\displaystyle \gamma }$ denotes Euler's constant.

Two related conjectures (see Commments) are

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

which is weaker. And,

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

which is stronger.

Currently, Firoozbakht's conjecture is included on the List of prime conjectures involving primes numbers listed from highest to least importance.[11] This short list of prime conjectures also includes Riemann hypothesis, Goldbach's conjecture, and Twin prime conjecture among others.

## Notes

1. ^ Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185.
2. ^ a b Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
3. ^ Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. pp. 34–42.
4. ^ Gaps between consecutive primes
5. ^ Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", arXiv.org > math > arXiv:1010.1399: 1–10.
6. ^ Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture".
7. ^ Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation, American Mathematical Society, 18 (88): 646–651, doi:10.2307/2002951, JSTOR 2002951, Zbl 0128.04203.
8. ^ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28.
9. ^ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399.
10. ^ Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471
11. ^ Sloane, Neil. "List of prime conjectures".

## References

• Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. ISBN 978-1479765584.
• Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. ISBN 0-387-20169-6.
• Riesel, Hans (1985). Prime Numbers and Computer Methods for Factorization, Second Edition. Birkhauser. ISBN 3-7643-3291-3.