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 in 1982.

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

Notes

1. Ribenboim, Paulo (2004). The Little Book of Bigger Primes (Second ed.). Springer-Verlag. p. 185. ISBN 978-0-387-20169-6.
2. Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
3. Oliveira e Silva, Tomás (December 30, 2015). "Gaps between consecutive primes". Retrieved 2024-11-01.
4. Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture".
5. Visser, Matt (August 2019). "Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap". Mathematics. 7 (8) 691. arXiv:1904.00499. doi:10.3390/math7080691.
6. Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", arXiv:1010.1399 [math.NT].
7. 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.
8. 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.
9. 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.
10. 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
11. Adleman, Leonard M.; McCurley, Kevin S. (1994), "Open problems in number-theoretic complexity. II", in Adleman, Leonard M.; Huang, Ming-Deh (eds.), Algorithmic Number Theory: Proceedings of the First International Symposium (ANTS-I) held at Cornell University, Ithaca, New York, May 6–9, 1994, Lecture Notes in Computer Science, vol. 877, Berlin: Springer, pp. 291–322, doi:10.1007/3-540-58691-1_70, ISBN 3-540-58691-1, MR 1322733
12. 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
13. Rivera, Carlos (2016). "Conjecture 78: P_n^(P_n+1/P_n)^n<=n^P_n". PrimePuzzles.net. Retrieved 2024-11-01.
14. Farhadian, Reza (October 2017). "On a New Inequality Related to Consecutive Primes". Acta Universitatis Danubius. Œconomica. 13 (5): 236–242. Archived from the original on 2018-04-19. Retrieved 2024-11-05.

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.