Pairs of square-free values of the type $\mathbf{n^2+1}$, $\mathbf{n^2+2}$

Abstract: In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive integer.