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

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 $x^2-D{y}^2=1$ 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.

28 pages. The proof of the theorem about consecutive k-free numbers has been reworked and errors and typos have been corrected. The error exponent is now much stronger for k>3. A further application of the method about the size of the fundamental solution of a Pell equation has been added. References to work of Dietmann/Marmon and Fouvry/Jouve have been added. The paper has been restructured

References in corpus (2)

• The density of twins of $k$-free numbers
• Square-free values of $n^2+1$

Cited by in corpus (12)

• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
• The density of twins of $k$-free numbers
• Height of rational points on quadratic twists of a given elliptic curve
• On the number of pairs of positive integers $\mathbf{x},\mathbf{y}\le \mathbf{H}$ such that ${\mathbf{x}}^2+{\mathbf{y}}^2+1$, ${\mathbf{x}}^2+{\mathbf{y}}^2+2$ are square-free
• Consecutive square-free values of the form $[\alpha \mathbf{p}],[\alpha \mathbf{p}]+1$
• Consecutive square-free numbers of a special form
• Moments of the distribution of $k$-free numbers in short intervals and arithmetic progressions
• Distance between natural numbers based on their prime signature
• Configurations Of Consecutive Primitive Roots
• Square-full values of quadratic polynomials
• On the $k$-free values of the polynomial $x{y}^k+C$
• Pairs of square-free values of the type ${\mathbf{n}}^2+1$, ${\mathbf{n}}^2+2$