On a Diophantine problem with one prime, two squares of primes and $s$ powers of two

Abstract

We refine a result of W.P. Li and Wang on the values of the form $ \lambda_1p_1 + \lambda_2p_2^{2} + \lambda_3p_3^{2} + \mu_1 2^{m_1} +...+ \mu_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers, $\lambda_1,\lambda_2,\lambda_{3}$ are nonzero real numbers, not all of the same sign,$\lambda_2 / \lambda_3$ is irrational and $\lambda_i/\mu_i \in \Q$, for $i\in\{1,2,3\}$.