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 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 < pk1+1/k;
k <
ln pk
ln pk+1 − ln pk.
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 pk up to 264
≈ 1.84×1019.
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 < 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 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
[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.
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