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 (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
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
Two related conjectures (see the comments of OEIS: A182514) are
which is weaker, and
which is stronger.
See also
- Prime number theorem
- Andrica's conjecture
- Legendre's conjecture
- Oppermann's conjecture
- Cramér's conjecture
Notes
- 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".
- ].
- Zbl 1390.11105.
- Zbl 0833.01018, archived from the original(PDF) on 2016-05-02.
- Zbl 0843.11043.
- Zbl 1226.11096
- MR 1322733
- 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.