Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap
arXiv:1904.00499 · doi:10.3390/math7080691
Abstract
The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps . \[ g_n \leq p_n \left(p_n^{1/n} -1 \right)\qquad\qquad\qquad (n \geq 1; \; Firoozbakht). \] \[ g_n \leq p_n \left((n\ln n)^{1/n} -1 \right)\qquad\qquad (n>4; \; Nicholson). \] \[ g_n \leq p_n \left( \left(p_n {\ln n\over\ln p_n}\right)^{1/n} -1 \right)\qquad (n>4; \; Farhadian). \] While a general proof of any of these conjectures is far out of reach I shall show that all three of these conjectures are unconditionally and explicitly verified for all primes below the location of the 81 maximal prime gap, certainly for all primes . For the Firoozbakht conjecture this is a very minor improvement on currently known results, for the Nicholson and Farhadian conjectures this may be more interesting.
V1: 9 pages; V2: now 10 pages. Updated to reflect the fact that an exhaustive search of prime gaps has (as of September 2018) verified that the 81st maximal prime gap occurs at some p > 2^{64}. Minor changes of presentation, typos fixed. Two references added