Journal of Integer Sequences, Vol. 18 (2015), Article 15.7.3

Primes in Intersections of Beatty Sequences

Glyn Harman
Department of Mathematics
Royal Holloway, University of London
Surrey TW20 0EX
United Kingdom


In this note we consider the question of whether there are infinitely many primes in the intersection of two or more Beatty sequences ⌊ ξjn + ηj⌋, nN, j = 1,...,k. We begin with a straightforward sufficient condition for a set of Beatty sequences to contain infinitely many primes in their intersection. We then consider two sequences when one ξj is rational. However, the main result we establish concerns the intersection of two Beatty sequences with irrational ξj. We show that, subject to a natural "compatibility" condition, if the intersection contains more than one element, then it contains infinitely many primes. Finally, we supply a definitive answer when the compatibility condition fails.

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Received May 26 2015; revised version received July 4 2015. Published in Journal of Integer Sequences, July 4 2015.

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