On a Diophantine problem with two primes and s powers of two

TL;DR

This paper refines a recent result on representing numbers as a combination of two primes and multiple powers of two, under specific irrational and rational ratio conditions.

Contribution

It improves previous bounds and extends the understanding of representing integers through primes and powers of two with particular ratio constraints.

Findings

Enhanced bounds on the representation problem.

Extended the class of numbers representable under given conditions.

Refined techniques for analyzing prime and power of two combinations.

Abstract

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is negative and irrational and $\lambda_1 / \mu_1$, $\lambda_2/\mu_2 \in \Q$.