Pairs of k-free Numbers, consecutive square-full Numbers

TL;DR

This paper improves error estimates for counting pairs of k-free and square-full numbers, extends results to r-tuples, and analyzes fundamental solutions of Pell equations using the approximate determinant method.

Contribution

It provides sharper error bounds for counting k-free and square-full number tuples and applies the approximate determinant method to Pell equations.

Findings

Improved error term for pairs of k-free integers.

Extended results to r-tuples of k-free numbers.

Established bounds for fundamental solutions of Pell equations.

Abstract

We consider the error term of the asymptotic formula for the number of pairs of $k$-free integers up to $x$. Our error term improves results by Heath-Brown, Brandes and Dietmann/Marmon. We then extend our results to $r$-tuples of $k$-free numbers and improve previous results by Tsang. Furthermore, we establish an error term for consecutive square-full integers. Finally, we will show that for all $\theta<3$ and for almost all $D$, the fundamental solution $\epsilon_D$ associated to the Pell equation $x^2-Dy^2=1$ satisfies $\epsilon_D> D^\theta$. This improves/recovers previous results by Fouvry and Jouve. The main tool of our work is the approximate determinant method.