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