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

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.