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

TL;DR

This paper refines previous results on representing numbers as a sum involving one prime, two prime squares, and multiple powers of two, under specific irrationality and rationality conditions on the coefficients.

Contribution

It improves upon earlier work by Li and Wang, providing a more precise analysis of the Diophantine problem with mixed prime and power-of-two terms under new coefficient constraints.

Findings

Extended the range of representable numbers in the given form.

Established conditions under which the representation is possible.

Enhanced understanding of the interplay between primes, prime squares, and 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 $i\in\{1,2,3\}$.