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

Abstract

Let $1<k<14/5$, ${\lambda }_1,{\lambda }_2,{\lambda }_3$ and ${\lambda }_4$ be non-zero real numbers, not all of the same sign such that ${\lambda }_1/{\lambda }_2$ is irrational and let $\omega$ be a real number. We prove that the inequality $|{\lambda }_1{p}_1+{\lambda }_2{p}_2^2+{\lambda }_3{p}_3^2+{\lambda }_4{p}_4^k-\omega |\le ({\max }_j{p}_j{)}^{-\frac{14-5k}{28k}+\epsilon }$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\epsilon >0$.

arXiv commentThis version contains a stronger result of the main theorem than the previous versions

References in corpus (3)

• On a ternary Diophantine problem with mixed powers of primes
• A Diophantine problem with prime variables
• On a Diophantine problem with one prime, two squares of primes and $s$ powers of two