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 [math]\displaystyle{ p_{n}^{1/n} }[/math] (where [math]\displaystyle{ p_n }[/math] is the nth prime) is a strictly decreasing function of n, i.e.,
- [math]\displaystyle{ \sqrt[n+1]{p_{n+1}} \lt \sqrt[n]{p_n} \qquad \text{ for all } n \ge 1. }[/math]
Equivalently:
- [math]\displaystyle{ p_{n+1} \lt p_n^{1+\frac{1}{n}} \qquad \text{ for all } n \ge 1, }[/math]
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 [math]\displaystyle{ g_n = p_{n+1} - p_n }[/math] would satisfy:[5]
- [math]\displaystyle{ g_n \lt (\log p_n)^2 - \log p_n \qquad \text{ for all } n \gt 4. }[/math]
Moreover:[6]
- [math]\displaystyle{ g_n \lt (\log p_n)^2 - \log p_n - 1 \qquad \text{ for all } n \gt 9, }[/math]
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
- [math]\displaystyle{ g_n \gt \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2 \approx 1.1229(\log p_n)^2 , }[/math]
occurs infinitely often for any [math]\displaystyle{ \varepsilon\gt 0, }[/math] where [math]\displaystyle{ \gamma }[/math] denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of OEIS: A182514) are
- [math]\displaystyle{ \left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n \lt e, }[/math]
which is weaker, and
- [math]\displaystyle{ \left(\frac{p_{n+1}}{p_n}\right)^n \lt n\log(n)\qquad \text{ for all } n \gt 5, }[/math]
which is stronger.
See also
- Prime number theorem
- Andrica's conjecture
- Legendre's conjecture
- Oppermann's conjecture
- Cramér's conjecture
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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