Prime numbers with Beatty sequences

William D. Banks ¹   ; Igor E. Shparlinski ²

¹ Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.
² Department of Computing Macquarie University Sydney, NSW 2109, Australia

Colloquium Mathematicum, Tome 115 (2009) no. 2, pp. 147-157

Résumé

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form $p=2\lfloor \alpha n\rfloor +1$, where $1 \alpha 2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q\lfloor \alpha n+\beta \rfloor +a$ with $n\leq N$, where $\alpha ,\beta $ are real numbers such that $\alpha $ is positive and irrational of finite type (which is true for almost all $\alpha $) and $a,q$ are integers with $0\leq a q\leq N^\kappa $ and $ \mathop {\rm gcd}(a,q)=1$, where $\kappa >0$ depends only on $\alpha $. We also prove a similar result for primes $p=\lfloor \alpha n+\beta \rfloor $ such that $p\equiv a\ ({\rm mod}\ q)$.

Détail

DOI : 10.4064/cm115-2-1

Keywords: study certain hamiltonian systems has led long conjecture existence infinitely many primes which form lfloor alpha rfloor where alpha fixed irrational number argument ribenboim coupled classical results about distribution fractional parts irrational multiples primes arithmetic progression immediately implies conjecture holds much precise asymptotic form motivated observation asymptotic formula number primes lfloor alpha beta rfloor leq where alpha beta real numbers alpha positive irrational finite type which almost alpha integers leq leq kappa mathop gcd where kappa depends only alpha prove similar result primes lfloor alpha beta rfloor equiv mod