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arXiv · 1904.00499

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

Matt Visser
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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.

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Matt Visser. 2019-03-31. Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap. https://doi.org/10.3390/math7080691

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