The Firoozbakht Conjecture

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This conjecture was first stated by the mathematician Farideh Firoozbakht from the University of Isfahan. It appeared in print in The Little Book of Bigger Primes by Paulo Ribenboim [3, page 185]. The Firoozbakht conjecture is one of the strongest upper bounds for prime gaps – somewhat stronger than the Cramér and Shanks conjectures. (Cramér predicted that the gaps near p are at most about as large as ln²p; moreover, Shanks [4] conjectured the asymptotic equality g ~ ln²p for maximal prime gaps g.)

The Firoozbakht conjecture.
Let pk be the k-th prime, then the sequence (pk)1/k is strictly decreasing.
Equivalent statements(pk)k+1 > (pk+1)k;   pk+1 < (pk)1+1/k;   k < ln pk ln pk+1 − ln pk.

As of 2015, a rigorous proof of the conjecture is not known – nor do we have any counterexamples. The conjecture is true for all pk < 4×1018  [1, Sect.3-4].
The conjecture implies: pk+1 pk  <  ln²pk − ln pk − 1 for k > 9  [2, Theorem 1].

Two ways to verify the Firoozbakht conjecture for all pk < 4×1018:
•  Using "safe bounds" and the table of first-occurrence prime gaps; see [1].
•  Using the sufficient condition below and the table of maximal prime gaps; see [2, Remark (i) on page 5].

Sufficient condition for the Firoozbakht conjecture:
If pk+1 pk  <  ln²pk − ln pk − 1.17 for all k > 9, then Firoozbakht’s conjecture is true [2, Theorem 3]. Because ln²pk − ln pk − 1.17 is an increasing function of pk, it is enough to check this condition only for maximal prime gaps, starting with the 5th maximal gap, i.e. for pk = A002386(n) ≥ 89. (The first four maximal gaps correspond to k ≤ 9.) Checking the conjecture for small primes pk ≤ 89 is easy with the table below.

[1] A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, Int. Math. Forum 10 (2015), 283-288. arXiv:1503.01744
[2] A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht’s conjecture, Journal of Integer Sequences 18 (2015), Article 15.11.2. arXiv:1506.03042
[3] P. Ribenboim, The Little Book of Bigger Primes, New York, Springer, 2004.
[4] D. Shanks, On maximal gaps between successive primes, Math. Comp. 18 (88) (1964), 646-651.

Table: A partial computational check of the Firoozbakht conjecture.
(See also verification up to 1000000 and verification up to 4×1018 via safe bounds.)

      k       p      p1/k    OK/fail  Alternative formulation:

See also:
Verification for primes up to one million (106).
Verification for primes up to four quintillion (4×1018).
Firoozbakht conjecture vs Cramér conjecture.

Copyright © 2011-2015, Alexei Kourbatov,