# Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap

## TL;DR

This paper verifies three prime gap conjectures up to the 81st maximal prime gap, covering all primes below 2^64, providing partial validation and extending current known results in prime number theory.

## Contribution

It unconditionally verifies the Firoozbakht, Nicholson, and Farhadian conjectures for all primes below the 81st maximal prime gap, extending the known range of these conjectures.

## Key findings

- All three conjectures verified below the 81st maximal prime gap.
- Verification covers all primes less than 2^64.
- Results provide partial validation for these prime gap bounds.

## 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.