Representations of integers as sums of primes from a Beatty sequence

Abstract

We study the problem of representing integers N ≡ κ (mod 2) as a sum of κ prime numbers from the Beatty sequencewhere α, β ∈ R with α > 1, and α is irrational and of finite type.In particular, we show that for κ = 2, almost all even numbers have such a representation if and only if α < 2, and for any fixed integer κ 3, all sufficiently large numbers N ≡ κ (mod 2) have such a representation if and only if α < κ.

Beatty sequences and prime numbers with restrictions on strongly q-additive functions

“…(see [2]) proved that if k ≥ 3, then every sufficiently large integer n ≡ k(mod 2) can be expressed as the sum of k primes from the sequence E α,β , provided that α < k and α has a finite type [see below (2.8)]. In their closing remarks, the authors of [3] note that their method can be used to extend the results of [3] to representations of an integer n in the form…”

On sums of primes from Beatty sequences

“…Over the years, a number of authors have studied variants of the three primes theorem with prime numbers restricted to various sequences of arithmetic interest. For instance, a recent work by Banks, Güloglu and Nevans [1] studies the question of representing integers as sums of primes from a Beatty sequence. Suppose that α and β are real numbers, with α > 1 and irrational.…”

“…Banks et al proved that if k ≥ 3, then every sufficiently large integer n ≡ k (mod 2) can be expressed as the sum of k primes from the sequence B α,β , provided that α < k and α "has a finite type" (see below). In their closing remarks, the authors of [1] note that their method can be used to extend the main results of [1] to representations of an integer n in the form (1)…”

“…. , α k of irrational numbers greater than 1, it appears to be much more difficult to estimate the number of representations of n ≡ k (mod 2) in the form (1), where p i lies in the Beatty sequence B α i ,β i ." The main purpose of the present note is to address the latter question in the case when at least one of the ratios α i /α j , 1 ≤ i, j ≤ k, is irrational.…”

Prime numbers with Beatty sequences

“…We remark that some additive problems of Goldbach type involving such primes have also been successfully studied; see [5,24,27]. Unfortunately, it seems that these results and techniques are not well known among non-experts (as we have mentioned, [18] gives a proof of a weaker result), so in part our motivation comes from a desire to make these techniques better known and to examine their potential and limitations.…”