2007/08/07 by William D. Banks, Banks, William D., Igor E. Shparlinski +1
Computer Science · Mathematics · #11B83 #11N13 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11B83 #msc:11N13 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0708.1015
arxiv created 2007/08/07 · openalex publication_date 2007/08/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes 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 imply 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\pmod q.