In number theory, Firoozbakht’s conjecture (or the Firoozbakht conjecture) 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 (where is the nth prime) is a strictly decreasing function of n, i.e.,
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×1012. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below .
see also A111943. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz and of Maier which suggest that
occurs infinitely often for any where denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of A182514) are
which is weaker, and
which is stronger.
- Prime number theorem
- Andrica's conjecture
- Legendre's conjecture
- Oppermann's conjecture
- Cramér's conjecture
- Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185.
- Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
- Gaps between consecutive primes
- Kourbatov, Alexei. "Prime Gaps: Firoozbakht Conjecture".
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- 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