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.

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}}}\qquad {\text{ for all }}n\geq 1,

see A182134, 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 $10^{{19}}$ .^[3]^[4]

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

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

Moreover:^[6]

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

see also 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

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.

Two related conjectures (see the comments of 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)\qquad {\text{ for all }}n>5,

which is stronger.

Notes

↑ Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185.
1 2 Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
↑ Gaps between consecutive primes
1 2 Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture".
↑ Sinha, Nilotpal Kanti (2010), On a new property of primes that leads to a generalization of Cramer's conjecture, pp. 1–10, arXiv:1010.1399, Bibcode:2010arXiv1010.1399K .
↑ 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, Bibcode:2015arXiv150603042K .
↑ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28 .
↑ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399 .
↑ Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471
↑ Leonard Adleman and Kevin McCurley, Open Problems in Number Theoretic Complexity, II. Algorithmic number theory (Ithaca, NY, 1994), 291–322, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.
↑ 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.

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[Ribenboim-1] Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185.

[ppagc30-2] 1 2 Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.

[3] Gaps between consecutive primes

[Kourbatov2018-4] 1 2 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, pp. 1–10, arXiv:1010.1399, Bibcode:2010arXiv1010.1399K .

[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, Bibcode:2015arXiv150603042K .

[7] Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28 .

[8] Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399 .

[Pintz07-9] Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471

[10] Leonard Adleman and Kevin McCurley, Open Problems in Number Theoretic Complexity, II. Algorithmic number theory (Ithaca, NY, 1994), 291–322, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.

[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

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Firoozbakht's conjecture

See also

Notes

References