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

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 ${ϵ}_D$ associated to the Pell equation $x^2-D{y}^2=1$ satisfies ${ϵ}_D>D^{\theta }$. This improves/recovers previous results by Fouvry and Jouve. The main tool of our work is the approximate determinant method.