Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Matt Visser
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Matt Visser: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand

Mathematics, 2018, vol. 6, issue 12, 1-7

Abstract: The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀ n ≥ 1 , is p n + 1 − p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 / ln p n + 1 − p n / ln p n < 11 / 25 ; ( n ≥ 1 ) . Then, by considering more general m t h roots, again assuming the Riemann hypothesis, I show that p n + 1 m − p n m < 44 / ( 25 e [ m − 2 ] ) ; ( n ≥ 3 ; m > 2 ) . In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln 2 p n + 1 − ln 2 p n < 9 ; ln 3 p n + 1 − ln 3 p n < 52 ; ln 4 p n + 1 − ln 4 p n < 991 ; ( n ≥ 1 ) . I shall also update the region on which Andrica’s conjecture is unconditionally verified.

Keywords: primes; prime gaps; Andrica’s conjecture; Riemann hypothesis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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