Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 297-310, 2023

Consecutive square-free values of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$

Ya-Fang Feng

Received April 8, 2022. Published online November 15, 2022.

Abstract: We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$. We also establish an asymptotic formula for $1\leq x, y, z \leq H$ such that $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$ are square-free. The method we used in this paper is due to Tolev.

Keywords: square-free number; Salié sum; Gauss sum

Classification MSC: 11N37, 11L05, 11L40

DOI: 10.21136/CMJ.2022.0154-22

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Ya-Fang Feng, School of Mathematical Sciences, Nanjing Normal University, No. 22, Hankou Road, Nanjing 210023, P. R. China, e-mail: yafangf@126.com

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