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.[3]

The conjecture states that p_{n}^{1/n}\, (where p_n\, is the nth prime) is a strictly decreasing function of n, i.e.,

p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1.

Equivalently: p_{n+1} < p_n^{1+\frac{1}{n}}\text{ for all } n \ge 1, see OEISA182134, OEISA246782.

By using table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×1012.[2]

If the conjecture is true, then the prime gap function g_n = p_{n+1} - p_n satisfies  g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4. [4] This is the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than Cramer-Shanks conjecture.[5] Moreover, Firoozbakht's conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture,[6] but inconsistent with the heuristics of Granville and Pintz[7][8][9] which suggest that  g_n > \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2 infinitely often for any \varepsilon>0.

Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 4×1018.[10]

Currently, Firoozbakht's conjecture is included on the List of prime conjectures involving primes numbers listed from highest to least importance.[11] This short list of prime conjectures also includes Riemann hypothesis, Goldbach's conjecture, and Twin prime conjecture among others.

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. ^ Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. pp. 34–42. 
  4. ^ Sinha, Nilotpal Kanti (2010), "On a new property of primes that leads to a generalization of Cramer's conjecture", > math > arXiv:1010.1399: 1–10 .
  5. ^ Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture". 
  6. ^ Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation (American Mathematical Society) 18 (88): 646–651, doi:10.2307/2002951, JSTOR 2002951, Zbl 0128.04203 .
  7. ^ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers", Scandinavian Actuarial Journal 1: 12–28 .
  8. ^ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers", 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 
  10. ^ Gaps between consecutive primes
  11. ^ Sloane, Neil. "List of prime conjectures". 


  • Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann's Hypothesis. Xlibris. ISBN 978-1479765584. 
  • 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.