On sums of primes from Beatty sequences

TL;DR

This paper investigates the representation of integers as sums of primes drawn from multiple Beatty sequences, extending classical additive prime problems to sequences defined by irrational parameters.

Contribution

It introduces a novel framework for analyzing sums of primes from Beatty sequences with irrational parameters, generalizing previous results on prime sums.

Findings

Established conditions for the representation of integers as sums of primes from Beatty sequences.

Extended classical additive prime results to sequences defined by irrational parameters.

Provided new insights into the distribution of primes within Beatty sequences.

Abstract

Let $k \ge 2$ and $\alpha_1, \beta_1, ..., \alpha_k, \beta_k$ be reals such that the $\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\alpha_i/\alpha_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \mathcal B_i = \left\{n \in \mathbb N : n = [ \alpha_i m + \beta_i ] \text{for some} m \in \mathbb Z \right\}. $$