  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.
