Prime numbers with Beatty sequences

William D. Banks, and Igor E. Shparlinski. "Prime numbers with Beatty sequences." Colloquium Mathematicae 115.2 (2009): 147-157. <http://eudml.org/doc/283554>.

@article{WilliamD2009,
abstract = {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⌊αn⌋ + 1, where 1 < α < 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⌊αn + β⌋ + a with n ≤ N, where α,β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a,q are integers with $0 ≤ a < q ≤ N^κ$ and gcd(a,q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a(mod q).},
author = {William D. Banks, Igor E. Shparlinski},
journal = {Colloquium Mathematicae},
keywords = {Beatty sequences; primes in arithmetic progressions},
language = {eng},
number = {2},
pages = {147-157},
title = {Prime numbers with Beatty sequences},
url = {http://eudml.org/doc/283554},
volume = {115},
year = {2009},
}

TY - JOUR
AU - William D. Banks
AU - Igor E. Shparlinski
TI - Prime numbers with Beatty sequences
JO - Colloquium Mathematicae
PY - 2009
VL - 115
IS - 2
SP - 147
EP - 157
AB - 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⌊αn⌋ + 1, where 1 < α < 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⌊αn + β⌋ + a with n ≤ N, where α,β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a,q are integers with $0 ≤ a < q ≤ N^κ$ and gcd(a,q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a(mod q).
LA - eng
KW - Beatty sequences; primes in arithmetic progressions
UR - http://eudml.org/doc/283554
ER -