Firoozbakht's conjecture

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Prime gap function

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 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.,

Equivalently: see OEISA182134, OEISA246782.

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 4×1018.[3]

If the conjecture is true, then the prime gap function satisfies [4] and, moreover, ;[5] see also OEISA111943. This is one of the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures.[6] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz[7][8][9] which suggest that infinitely often for any where denotes the Euler–Mascheroni constant.

Two related conjectures (see the comments of OEISA182514) are


which is weaker, and

for all values with ,

which is stronger.

See also[edit]


  1. ^ Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185. 
  2. ^ a b Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012. 
  3. ^ Gaps between consecutive primes
  4. ^ Sinha, Nilotpal Kanti (2010), On a new property of primes that leads to a generalization of Cramer's conjecture, pp. 1–10, arXiv:1010.1399free to read .
  5. ^ Kourbatov, Alexei (2015), "Upper bounds for prime gaps related to Firoozbakht's conjecture", Journal of Integer Sequences, 18 (Article 15.11.2), arXiv:1506.03042free to read .
  6. ^ Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture". 
  7. ^ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28 .
  8. ^ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: 388–399 .
  9. ^ Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): 232–471 


  • 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.