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 kth 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}
ln_{ }p_{k+1} − ln_{ }p_{k}.
As of 2015, a rigorous proof of the conjecture is not known – nor do we have any counterexamples.
The conjecture is true for all p_{k} < 4×10^{18} [1, Sect.34].
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} < 4×10^{18}:
• Using "safe bounds" and the table of firstoccurrence 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’s 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), 283288. 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), 646651.
Table: A partial computational check of the Firoozbakht conjecture.
(See also
verification up to 1000000 and
verification up to 4×10^{18} via safe bounds.)
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.
