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**
Primes in Intersections of Beatty Sequences
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Glyn Harman

Department of Mathematics

Royal Holloway, University of London

EGHAM

Surrey TW20 0EX

United Kingdom

**Abstract:**

In this note we consider the question of whether there are infinitely
many primes in the intersection of two or more Beatty sequences
⌊ ξ_{j}*n* + η_{j}⌋,
*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 ξ_{j}
is rational. However, the
main result we establish concerns the intersection of two Beatty
sequences with irrational ξ_{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.

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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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