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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Wikipedia_primegaps.png/450px-Wikipedia_primegaps.png)
The conjecture states that (where is the nth prime) is a strictly decreasing function of n, i.e.,
Equivalently:
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 would satisfy:[5]
Moreover:[6]
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
occurs infinitely often for any where denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of OEIS: A182514) are
which is weaker, and
which is stronger.
See also
editNotes
edit- ^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185. ISBN 9780387201696.
- ^ a b Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
- ^ Gaps between consecutive primes
- ^ a b 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", 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), arXiv:1506.03042, MR 3436186, Zbl 1390.11105.
- ^ 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.
- ^ 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.
- ^ 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
- ^ 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
- ^ 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
edit- 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.