Checking the Firoozbakht conjecture: safe bounds
Doing Math with JavaScript
The table below gives "safe bounds" for the
Firoozbakht conjecture
as applied to prime gaps of a given size g ∈ [2,1510].
For example, a gap of size 22 is safe (it cannot violate the Firoozbakht conjecture)
when such a gap occurs between primes above 181.
The safe bounds are found by solving the inequalities (based on
Dusart's π(x) bounds):
0 < x/(ln x − 1.2) < ln x/(ln(x+g) − ln x)
for x > 4;
0 < x/(ln x − 1.1) < ln x/(ln(x+g) − ln x)
for x > 60184, g ≥ 110.
Use this data in conjunction with the table of first occurrence prime gaps
up to 10^{19}.
For g = 2 and g = 4, one can manually
check the Firoozbakht conjecture for small primes
below the respective safe bounds.
For g ∈ [6,1510], the actual first occurrence of prime gap g is already safe.
(For g ∈ [22,1510], even the number of digits in the safe bound
is smaller than that of the firstoccurrence primes with gap g.)
This verifies the Firoozbakht conjecture for primes up to 10^{19}
(because gaps g > 1510 never occur between primes below 10^{19}).
To obtain safe bounds for large gaps, you can use e.g.
Wolfram Alpha (the normal floatingpoint precision might not be sufficient).
See also arXiv:1503.01744,
arXiv:1506.03042.
Gap of size g ... is safe above (digits)
Note:
It is easy to check that the safe bounds form an increasing sequence (as a function of increasing gap sizes g.)
You might start checking actual firstoccurrence gap sizes versus safe bounds one by one.
However, once you encounter a safe maximal prime gap (such as the gap
g = 8 following the prime p_{k} = 89 –
which is above the respective safe bound, i.e. above 28),
any further verification can then be performed for maximal prime gaps only, i.e. for
g = A005250(n), with
p_{k} = A002386(n).

