Carmichael numbers composed of primes from a Beatty sequence

Abstract

Let α, β ∈ R be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence B α,β = (αn + β) ∞ n=1. We conjecture that the same result holds true when α is an irrational number of infinite type.

On primes in Beatty sequences

“…In this paper, we are interested in the von Mangoldt function over Beatty sequences in arithmetic sequences. Ideas we are involved can also be used to improve some related results in [3,4] with some special cases. The so-called Beatty sequence of integers defined by B α,β := {[αn + β]} ∞ n=1 , where α and β are fixed real numbers and [x] denotes the greatest integer not larger than x.…”

“…The analytic properties of Beatty sequences have been studied by many experts. For example, one can refer to [3,4] and the references therein. Specially, we focus on some such results appeared in [3,4].…”

“…For example, one can refer to [3,4] and the references therein. Specially, we focus on some such results appeared in [3,4]. Such type results have some relations with a result of Jia [6] and a conjecture of Long [9] and draw many authors special attention.…”

“…where the implied constant depends only on α and β. Recently, an explicit version of the constant κ was given in [4]. In [4], Banks and Yeager showed the follows.…”

Carmichael numbers and the sieve

“…The conjecture was proved unconditionally by Matomäki [20] in the special case that a is a quadratic residue modulo b, and using an extension of her methods Wright [23] established the conjecture in full generality. The techniques introduced in [1] have led to many other investigations into the arithmetic properties of Carmichael numbers; see [2][3][4][5]7,8,10,[14][15][16][17][18][19]21,24] and the references therein.…”