Geodesic
Consecutive square-free values of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 297-310.

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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.
DOI : 10.21136/CMJ.2022.0154-22
Classification : 11L05, 11L40, 11N37
Keywords: square-free number; Salié sum; Gauss sum 
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Feng, Ya-Fang. Consecutive square-free values of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 297-310. doi : 10.21136/CMJ.2022.0154-22. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0154-22/

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