On a ternary Diophantine problem with mixed powers of primes

TL;DR

This paper proves that for certain ranges of k, the inequality involving a linear combination of primes raised to different powers has infinitely many solutions in primes, extending the understanding of mixed power Diophantine problems.

Contribution

It establishes the existence of infinitely many prime solutions to a ternary Diophantine inequality with mixed prime powers for 1<k<33/29, a new result in the field.

Findings

Infinitely many solutions exist for the inequality with primes for 1<k<33/29.

The solutions satisfy a specific approximation bound involving the maximum prime.

The result applies under conditions on the coefficients and irrationality of their ratios.

Abstract

Let $1 < k < 33 / 29$. We prove that if $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real numbers, not all of the same sign and that $\lambda_1 / \lambda_2$ 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^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $ has infinitely many solutions in prime variables $p_1$, ..., $p_k$.