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

Abstract

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

References in corpus (2)

• Pairs of k-free Numbers, consecutive square-full Numbers
• Consecutive square-free values of the form $[\alpha \mathbf{p}],[\alpha \mathbf{p}]+1$

Cited by in corpus (1)

• Consecutive square-free values of the form $[\alpha \mathbf{p}],[\alpha \mathbf{p}]+1$