Moscow Mathematical Journal

Volume 25, Issue 4, October–December 2025 pp. 495–509.

Consecutive Square-Free Values for some Polynomials

Authors: S. I. Dimitrov (1)
Author institution:(1) Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Blvd. St. Kliment Ohridski 8, Sofia 1000, Bulgaria
Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 105, Sofia 1113, Bulgaria

Summary:

In this paper we study the distribution of consecutive square-free numbers of the forms $x^2+y^2+z+1$, $x^2+y^2+z+2$ and $x^2+y^2+z^2+z+1$, $x^2+y^2+z^2+z+2$, respectively. We establish asymptotic formulas for each of these two cases.

2020 Math. Subj. Class. 11L05, 11N25, 11N37.

Keywords: Square-free numbers, asymptotic formula, Gauss sum, Salié sum.

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