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In the present paper we show that there exist infinitely many consecutivesquare-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give anasymptotic formula for the number of pairs of positive integers $x, y \leq H$such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.

On the number of pairs of positive integers $\mathbf{x, y \leq H}$ such that $\mathbf{x^2+y^2+1}$, $\mathbf{x^2+y^2+2}$ are square-free