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 who stated it first 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:

${\displaystyle p_{n+1}<p_{n}^{1+{\frac {1}{n}}}\qquad {\text{ for all }}n\geq 1,}$

see OEIS:  A182134 , OEIS:  A246782 .

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×10¹². ^[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 2⁶⁴≈1.84×10¹⁹. ^{[3] [4]}

If the conjecture were true, then the prime gap function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ would satisfy: ^[5]

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

Moreover: ^[6]

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

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 ^{[7] [8] [9]} and of Maier ^{[10] [11]} 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.

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

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

which is weaker, and

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

which is stronger.

See also

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

Notes

1. Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition . Springer-Verlag. p.  185. ISBN   9780387201696.
2. Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture" . Retrieved 22 August 2012.
3. Gaps between consecutive primes
4. Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture".
5. Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", arXiv: 1010.1399 [math.NT].
6. Kourbatov, Alexei (2015), "Upper bounds for prime gaps related to Firoozbakht's conjecture", Journal of Integer Sequences, 18 (Article 15.11.2), arXiv: 1506.03042 , MR   3436186, Zbl   1390.11105 .
7. Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28, doi:10.1080/03461238.1995.10413946, MR   1349149, Zbl   0833.01018, archived from the original (PDF) on 2016-05-02.
8. Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399, doi:10.1007/978-3-0348-9078-6_32, ISBN   978-3-0348-9897-3, Zbl   0843.11043 .
9. Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471, doi: 10.7169/facm/1229619660 , MR   2363833, S2CID   120236707, Zbl   1226.11096
10. Leonard Adleman and Kevin McCurley, "Open Problems in Number Theoretic Complexity, II ^{[ permanent dead link ]}" (PS), Algorithmic number theory (Ithaca, NY, 1994), Lecture Notes in Comput. Sci. 877: 291–322, Springer, Berlin, 1994. doi : 10.1007/3-540-58691-1_70. ISBN   978-3-540-58691-3.
11. Maier, Helmut (1985), "Primes in short intervals", The Michigan Mathematical Journal, 32 (2): 221–225, doi: 10.1307/mmj/1029003189 , ISSN   0026-2285, MR   0783576, Zbl   0569.10023

References

• 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.