Primes in Intersections of Beatty Sequences
Department of Mathematics
Royal Holloway, University of London
Surrey TW20 0EX
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⌋,
n ∈ N, 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.
Full version: pdf,
May 26 2015;
revised version received July 4 2015.
Published in Journal of Integer Sequences, July 4 2015.
Journal of Integer Sequences home page