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

2012/12/13 by Thomas Reuss, T. Reuss, Reuss, T. · 1 citation
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1212.3150

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

openalex publication_date 2012/12/13 · arxiv created 2014/03/19 · arxiv updated 2014/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 x²-Dy²=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.

Cited by

• Twins of s-free numbers

Related

• Pairs of square-free values of the type n²+1, n²+2
• On the number of pairs of positive integers x, y ≤ H such that x²+y²+1, x²+y²+2 are square-free
• On the number of pairs of positive integers x, y ≤ H such that x² + y² + 1 is squarefree
• Integer solutions of Pell equation in bounded regions
• Three consecutive near-square squarefree numbers