Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap
Matt Visser
TL;DR
This work unconditionally verifies the Firoozbakht, Nicholson, and Farhadian conjectures for all primes below the location of the 81st maximal prime gap, in particular covering all primes with $p<2^{64}$. It derives monotone, $n$-dependent sufficient conditions on the prime gaps and propagates these bounds through intervals between known maximal gaps, using the inequality $e^x-1>x$ and bounds on $p_n$ to produce explicit, verifiable functions $f(n)$. The results explicitly confirm Firoozbakht and Nicholson for $p<2^{64}$ and, separately, Farhadian for the same range (excluding very small primes), thereby extending the verified domain beyond prior work. The methodology supports semi-automation and can adapt as new maximal prime gaps are discovered, contributing a rigorous, scalable approach to testing these conjectures in the absence of general proofs.
Abstract
The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps $g_n := p_{n+1}-p_n$. \[ g_n \leq p_n \left(p_n^{1/n} -1 \right)\qquad\qquad\qquad (n \geq 1; \; Firoozbakht). \] \[ g_n \leq p_n \left((n\ln n)^{1/n} -1 \right)\qquad\qquad (n>4; \; Nicholson). \] \[ g_n \leq p_n \left( \left(p_n {\ln n\over\ln p_n}\right)^{1/n} -1 \right)\qquad (n>4; \; Farhadian). \] 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$^{st}$ maximal prime gap, certainly for all primes $p <2^{64}$. 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.