On 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
Volume 194 / 2020
Acta Arithmetica 194 (2020), 281-294
MSC: Primary 11L05, 11N25; Secondary 11N37.
DOI: 10.4064/aa190118-25-7
Published online: 6 March 2020
Abstract
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 establish an asymptotic 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.
