A Diophantine problem with a prime and three squares of primes

TL;DR

This paper proves that a specific Diophantine inequality involving a prime and three squares of primes has infinitely many solutions under certain irrationality and sign conditions on the coefficients.

Contribution

It establishes the existence of infinitely many solutions to a Diophantine inequality with a prime and three prime squares, extending previous results in prime number theory.

Findings

Infinitely many solutions exist for the inequality with specified conditions.

The solutions involve primes and their squares satisfying the inequality.

The bound on the inequality is explicitly given as a function of the maximum prime.

Abstract

We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number then, for any $\eps > 0$ the inequality $ \bigl|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \lambda_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $ has infinitely many solution in prime variables $p_1$, ..., $p_4$