Firoozbakht's conjecture

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 $p_{n}^{1 / n}$ (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}} for all n \geq 1 .

Equivalently:

p_{n + 1} < p_{n}^{1 + \frac{1}{n}} 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 $g_{n} = p_{n + 1} - p_{n}$ would satisfy:^[5]

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

Moreover:^[6]

g_{n} < (\log p_{n})^{2} - \log p_{n} - 1 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

g_{n} > \frac{2 - ε}{e^{γ}} (\log p_{n})^{2} \approx 1.1229 (\log p_{n})^{2},

occurs infinitely often for any $ε > 0,$ where $γ$ denotes the Euler–Mascheroni constant.

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

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

which is weaker, and

{(\frac{p_{n + 1}}{p_{n}})}^{n} < n \log (n) for all n > 5,

which is stronger.

Notes

↑ Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185. ISBN 9780387201696. https://archive.org/details/littlebookbigger00ribe_610.
↑ ^2.0 ^2.1 Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". http://www.primepuzzles.net/conjectures/conj_030.htm.
↑ Gaps between consecutive primes
↑ ^4.0 ^4.1 Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture". http://www.javascripter.net/math/primes/firoozbakhtconjecture.htm.
↑ Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", arXiv:1010.1399 [math.NT].
↑ Kourbatov, Alexei (2015), "Upper bounds for prime gaps related to Firoozbakht's conjecture", Journal of Integer Sequences 18 (Article 15.11.2), http://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html .
↑ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers", Scandinavian Actuarial Journal 1: 12–28, doi:10.1080/03461238.1995.10413946, http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf .
↑ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers", 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, http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf .
↑ 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, http://projecteuclid.org/euclid.facm/1229619660
↑ Leonard Adleman and Kevin McCurley, "Open Problems in Number Theoretic Complexity, II" (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.
↑ Maier, Helmut (1985), "Primes in short intervals", The Michigan Mathematical Journal 32 (2): 221–225, doi:10.1307/mmj/1029003189, ISSN 0026-2285, http://projecteuclid.org/euclid.mmj/1029003189

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 (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185. ISBN 9780387201696. https://archive.org/details/littlebookbigger00ribe_610.

[ppagc30-2] 2.0 ^2.1 Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". http://www.primepuzzles.net/conjectures/conj_030.htm.

[3] Gaps between consecutive primes

[Kourbatov2018-4] 4.0 ^4.1 Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture". http://www.javascripter.net/math/primes/firoozbakhtconjecture.htm.

[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), http://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html .

[7] Granville, A. (1995), "Harald Cramér and the distribution of prime numbers", Scandinavian Actuarial Journal 1: 12–28, doi:10.1080/03461238.1995.10413946, http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf .

[8] Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers", 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, http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf .

[Pintz07-9] 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, http://projecteuclid.org/euclid.facm/1229619660

[10] Leonard Adleman and Kevin McCurley, "Open Problems in Number Theoretic Complexity, II" (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, http://projecteuclid.org/euclid.mmj/1029003189

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