Prime numbers with Beatty sequences

• Autorzy: William D. Banks , Igor E. Shparlinski
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 11L03: Trigonometric and exponential sums, general
• 11B83: Special sequences and polynomials
• 11N13: Primes in progressions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 115
• Numer: 2
• Strony: 147-157

Daty

wydano

2009

Twórcy

autor

William D. Banks

• Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

autor

Igor E. Shparlinski

• Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

Identyfikatory

DOI

10.4064/cm115-2-1