A Diophantine inequality with five squares of Piatetski-Shapiro primes

TL;DR

This paper proves the existence of infinitely many prime quintuples of Piatetski-Shapiro primes satisfying a specific Diophantine inequality involving quadratic powers, extending to higher powers with similar results.

Contribution

It introduces new results on Diophantine inequalities with Piatetski-Shapiro primes involving quadratic, cubic, and quartic powers, demonstrating their infinite occurrence.

Findings

Infinitely many prime quintuples satisfy the inequality with quadratic powers.

Analogous results hold for cubic and quartic powers.

The inequalities involve Piatetski-Shapiro primes with fractional exponents.

Abstract

Let $[\,\cdot\,]$ denote the floor function. Assume that $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is a real number. Let $\frac{71}{72}<\gamma<1$ and $\theta>0$. We prove that there exist infinitely many quintuples of primes $p_1,\, p_2,\, p_3,\, p_4,\, p_5$ satisfying the Diophantine inequality \begin{equation*} \big|\lambda_1p^2_1 + \lambda_2p^2_2 + \lambda_3p^2_3+ \lambda_4p^2_4 + \lambda_5p^2_5+\eta\big|<\big(\max p_j\big)^{\frac{71-72\gamma}{29}+\theta}\,, \end{equation*} where $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3,\,4,\,5$. We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.