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

School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Mathematics 2018, 6(12), 289; https://doi.org/10.3390/math6120289
Received: 18 September 2018 / Revised: 18 November 2018 / Accepted: 26 November 2018 / Published: 27 November 2018
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. View Full-Text
Keywords: primes; prime gaps; Andrica’s conjecture; Riemann hypothesis primes; prime gaps; Andrica’s conjecture; Riemann hypothesis
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Visser, M. Variants on Andrica’s Conjecture with and without the Riemann Hypothesis. Mathematics 2018, 6, 289.

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