Square-Free Numbers of the Form $x^2+y^2+z^2+z+1$ and $x^2+y^2+z+1$

S. I. Dimitrov<sup>ab

a</sup> Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, 1756, Bulgaria
^b Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, 1113, Bulgaria

Abstract: In this paper we study the distribution of square-free positive integers of the form $x^2+y^2+z^2+z+1$ and $x^2+y^2+z+1$. We establish asymptotic formulas for the number of triples of positive integers $x, y, z \leq H$ such that $x^2+y^2+z^2+z+1$ is square-free and such that $x^2+y^2+z+1$ is square-free.

Keywords: square-free number, asymptotic formula, Gauss sum, Salié sum.

Received: 11.05.2023
Revised: 11.05.2023

Language: English

 English version: Mathematical Notes, 2023, 114:6, 1169–1183

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