Mathematics (Aug 2019)

Verifying the Firoozbakht, Nicholson, and Farhadian Conjectures up to the 81st Maximal Prime Gap

  • Matt Visser

DOI
https://doi.org/10.3390/math7080691
Journal volume & issue
Vol. 7, no. 8
p. 691

Abstract

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The Firoozbakht, Nicholson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps g n : = p n + 1 - p n . 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 as yet unknown location of the 81st maximal prime gap, certainly for all primes p < 2 64 . For the Firoozbakht conjecture itself this is a rather minor improvement on currently known results, but for the somewhat stronger Nicholson and Farhadian conjectures this may be considerably more interesting. Sequences: A005250 A002386 A005669 A000101 A107578 A246777 A246776.

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