A Diophantine inequality involving different powers of primes of the form $[n^c]$

TL;DR

This paper proves the existence of infinitely many prime triples satisfying a Diophantine inequality involving different powers of primes, with the primes approximated by fractional powers of integers, extending understanding of prime distributions in such inequalities.

Contribution

It establishes the infinitude of prime triples satisfying a specific Diophantine inequality with primes approximated by fractional powers, under certain irrationality and sign conditions.

Findings

Infinitely many prime triples satisfy the inequality.

Primes can be approximated by fractional powers of integers.

The inequality holds under specified irrationality and sign conditions.

Abstract

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let $\frac{219}{220}<\gamma<1$ and $\theta>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda_1p_1 + \lambda_2p_2 + \lambda_3p^4_3+\eta|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220\gamma}{208}+\theta} \end{equation*} and such that $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3$.