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 ${\lambda }_1{p}_1+{\lambda }_2{p}_2^2+{\lambda }_3{p}_3^2+{\mu }_1{2}^{m_1}+...+{\mu }_s{2}^{m_s},$ 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 $λ_i/μ_i \in \Q$, for $i\in \{1,2,3\}$.

arXiv commentimproved minor arc estimate

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• Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime