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 1019. 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 first-occurrence primes with gap g.) This verifies the Firoozbakht conjecture for primes up to 1019 (because gaps g > 1510 never occur between primes below 1019). To obtain safe bounds for large gaps, you can use e.g. Wolfram Alpha (the normal floating-point 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 first-occurrence 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 pk = 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 pk = A002386(n).

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