Geodesic
Primes in intersections of Beatty sequences
Journal of integer sequences, Tome 18 (2015) no. 7
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.
Classification : 11B83, 11N13, 11J71
Keywords: Beatty sequence, prime number 
@article{JIS_2015__18_7_a2,
     author = {Harman,  Glyn},
     title = {Primes in intersections of {Beatty} sequences},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {7},
     zbl = {1332.11068},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_7_a2/}
}
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Harman,  Glyn. Primes in intersections of Beatty sequences. Journal of integer sequences, Tome 18 (2015) no. 7. http://geodesic.mathdoc.fr/item/JIS_2015__18_7_a2/