J. Integer Seq. 18(7), Article 15.7.3, 12 p., electronic only (2015)
Summary
Summary: In this note we consider the question of whether there are infinitely many primes in the intersection of two or more Beatty sequences $\lfloor \xi _{j}n + \eta _{j}\rfloor , n \in $ N, $j = 1,\dots ,k$. 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 $\xi _{j}$ is rational. However, the main result we establish concerns the intersection of two Beatty sequences with irrational $\xi _{j}$. 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.
