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 p_k be the k-th prime, then the sequence (p_k)^1/k is strictly decreasing.
Equivalent statements: (p_k)^k+1> (p_k+1)^k; p_k+1 < p_k^1+1/k; k < ln p_k lnp_k+1 − lnp_k.

As of 2019, a rigorous proof of the conjecture is not known – nor do we have any counterexamples. The conjecture is true for all primes p_k up to 2⁶⁴ ≈ 1.84×10¹⁹.
The conjecture implies: p_k+1 − p_k < ln²p_k − ln p_k − 1 for k > 9 [2, Theorem 1].

Two ways to verify the Firoozbakht conjecture for all p_k < 2⁶⁴:
• 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 p_k+1 − p_k < ln²p_k − ln p_k − 1.17 for all k > 9, then Firoozbakht痴 conjecture is true [2, Theorem 3]. Because ln²p_k − ln p_k − 1.17 is an increasing function of p_k, it is enough to check this condition only for maximal prime gaps, starting with the 5th maximal gap, i.e. for p_k = A002386(n) ≥ 89. (The first four maximal gaps correspond to k ≤ 9.) Checking the conjecture for small primes p_k ≤ 89 is easy with the table below.

References
[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痴 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 10¹⁹ 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 ten quintillion (10¹⁹).
• Firoozbakht conjecture vs Cramér conjecture.