Prime numbers with beatty sequences

William D. Banks*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticle

20 Citations (Scopus)

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 ≤ Nκ 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).

Original languageEnglish
Pages (from-to)147-157
Number of pages11
JournalColloquium Mathematicum
Volume115
Issue number2
DOIs
Publication statusPublished - 2009

Keywords

  • Beatty sequences
  • prime numbers
  • exponential sums

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