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 CramérShanks 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 p_{k} be the kth prime, then the sequence
(p_{k})^{1/k} is strictly decreasing.
Alternative formulation:
(p_{k})^{k+1} > (p_{k+1})^{k},
where p_{k} is the kth prime.
As of 2014, a rigorous proof of the conjecture is not known – nor do we have any counterexamples.
The conjecture has been verified up to 4×10^{18}
(arXiv:1503.01744).
If the conjecture is true,
p_{k+1} − p_{k}
< ln²p_{k} − ln p_{k} − 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 4×10^{18}.)
k p p^{1/k} OK/fail Alternative formulation:
See also:
• Verification for primes up to one million (10^{6}).
• Verification for primes up to four quintillion (4×10^{18}).
• Firoozbakht conjecture vs Cramér conjecture.
