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 (predicting that the gaps near x are at most about as large as ln²x). The precise formulation is as follows:

The Firoozbakht conjecture.
Let pk be the k-th prime, then the sequence (pk)1/k is strictly decreasing.
Alternative formulation(pk)k+1 > (pk+1)k, where pk is the k-th prime.

As of 2014, a rigorous proof of the conjecture is not known – nor do we have any counterexamples. The conjecture is true for all pk < 4×1018 (arXiv:1503.01744). The conjecture implies: pk+1 pk  <  ln²pk − ln pk − 1 for k > 9 (arXiv:1506.03042).

Here is a partial computational check of the Firoozbakht conjecture.
(See also verification up to 1000000 and verification up to 4×1018.)

      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,