Firoozbakht's conjecture explained

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 in 1982.

The conjecture states that

(where is the nth prime) is a strictly decreasing function of n, i.e., } < \sqrt[n] \qquad \text n \ge 1.

Equivalently:

see, .

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444.^[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 2⁶⁴ ≈ .^{[3] [4] [5]}

If the conjecture were true, then the prime gap function

would satisfy:^[6]

Moreover:^[7]

see also . 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^{[8] [9]} and of Maier which suggest that

occurs infinitely often for any

where denotes the Euler–Mascheroni constant.

Three related conjectures (see the comments of) are variants of Firoozbakht's. Forgues notes that Firoozbakht's can be written

where the right hand side is the well-known expression which reaches Euler's number in the limit , suggesting the slightly weaker conjecture

Nicholson and Farhadian^{[10] [11]} give two stronger versions of Firoozbakht's conjecture which can be summarized as:

where the right-hand inequality is Firoozbakht's, the middle is Nicholson's (since ; see), and the left-hand inequality is Farhadian's (since ; see).

All have been verified to 2⁶⁴.^[5]

See also

• Prime number theorem
• Andrica's conjecture
• Legendre's conjecture
• Oppermann's conjecture
• Cramér's conjecture

References

• Book: Ribenboim, Paulo. The Little Book of Bigger Primes . Second . Springer-Verlag . 2004. 0-387-20169-6.
• Book: Riesel, Hans. Prime Numbers and Computer Methods for Factorization . Second . Birkhauser. 1985. 3-7643-3291-3.

Notes and References

1. Book: Ribenboim, Paulo. The Little Book of Bigger Primes . Second . 2004. limited. Springer-Verlag. 185. 978-0-387-20169-6 .
2. Web site: Rivera. Carlos. Conjecture 30. The Firoozbakht Conjecture. 22 August 2012.
3. Web site: Tomás . Oliveira e Silva . Gaps between consecutive primes . December 30, 2015 . 2024-11-01.
4. Web site: Kourbatov . Alexei . Prime Gaps: Firoozbakht Conjecture .
5. Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap . Matt . Visser . Matt Visser . 1904.00499 . 10.3390/math7080691 . free . Mathematics . August 2019 . 7 . 8 . 691.
6. Sinha . Nilotpal Kanti . On a new property of primes that leads to a generalization of Cramer's conjecture . 2010 . math.NT . 1010.1399. cs2. .
7. .
8. .
9. .
10. Web site: Carlos . Rivera . Conjecture 78: P_n^(P_n+1/P_n)^n<=n^P_n . 2016 . PrimePuzzles.net . 2024-11-01.
11. Reza . Farhadian. October 2017 . On a New Inequality Related to Consecutive Primes . Acta Universitatis Danubius. Œconomica . 13 . 5 . 236-242 . http://web.archive.org/web/20180419103716/http://journals.univ-danubius.ro/index.php/oeconomica/article/view/4223 . 2018-04-19 . dead.