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arXiv · 1212.3150

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

T. Reuss
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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 $θ<3$ and for almost all $D$, the fundamental solution $ε_D$ associated to the Pell equation $x^2-Dy^2=1$ satisfies $ε_D> D^θ$. This improves/recovers previous results by Fouvry and Jouve. The main tool of our work is the approximate determinant method.

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T. Reuss. 2014-03-19. Pairs of k-free Numbers, consecutive square-full Numbers. https://arxiv.org/abs/1212.3150

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