Consecutive primes and Beatty sequences

Fix irrational numbers alpha, alpha > 1 of finite type and real numbers beta,beta >= 0, and let B and B be the Beatty sequences B := ([alpha m + beta])(m is an element of N) and B := ([alpha m +beta])(m is an element of N). In this note, we study the distribution of pairs (p, p(#)) of consecutive primes for which p is an element of B and p(#) is an element of B. We conjecture that the estimate |{p <= x : p is an element of B and p# is an element of B}| = (alpha alpha)(-1)pi(x) + O(x(Iogx)(-3/2+epsilon)) holds for every fixed epsilon > 0, and we give a heuristic argument to support this prediction which relies (in part) on a strong form of the Hardy-Littlewood conjectures. (C) 2018 Elsevier Inc. All rights reserved.