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

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 $\lambda_1 / \mu_1$, $\lambda_2/\mu_2 \in \Q$.