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

TL;DR

This paper proves that for certain real coefficients and a range of k, there are infinitely many prime solutions to a Diophantine inequality involving one prime, two squares of primes, and a k-th power of a prime, with a specific approximation bound.

Contribution

It establishes the existence of infinitely many prime solutions to a new Diophantine inequality involving mixed prime powers for 1<k<14/5, extending previous approximation results.

Findings

Infinitely many solutions exist for the inequality with specified prime variables.

The approximation bound depends on the maximum of the prime variables and the parameter k.

The result holds for any small positive epsilon, indicating robustness of the approximation.

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 $|\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^k-\omega|\le (\max_j p_j)^{-\frac{14-5k}{28k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$.