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 (2004, page 185). The Firoozbakht conjecture is one of the strongest upper bounds for prime gaps – even somewhat stronger than the Cramér-Shanks conjecture. (Cramér predicted that the gaps near p are at most about as large as ln²p; moreover, Shanks conjectured the asymptotic equivalence 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.

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’s conjecture, Journal of Integer Sequences 18 (2015), Article 15.11.2. arXiv:1506.03042

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.

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