Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3

On Intervals (kn, (k + 1)n) Containing a Prime for All n > 1


Vladimir Shevelev
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105
Israel

Charles R. Greathouse IV
3214 Whitethorn Road
Cleveland Heights, OH 44118
USA

Peter J. C. Moses
Moparmatic Co.
1154 Evesham Road
Astwood Bank, Redditch, Worcestershire
B96 6DT England

Abstract:

We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes 1, 2, 3, 5, 9, 14 and no others, at least for k ≤ 100,000,000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n > Nk(m), then the interval (kn, (k + 1)n) contains at least m primes.

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(Concerned with sequences A084140 A104272 A164952 A185004 A185005 A185006 A185007 A218769 A218831 A218850 A220268 A220269 A220273 A220274 A220281 A220293 A220462 A220463 A220474 A220475.)

Received January 1 2013; revised version received May 7 2013; June 10 2013; July 25 2013. Published in Journal of Integer Sequences, July 31 2013.

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