Apollonian circle packings: number theory. II: Spherical and hyperbolic packings (Q2479870)

In this paper, the authors develop a reduction step for the orbit of the so called Apollonian group which acts on the integral solutions of Diophantine equation \[ 2(w^2+x^2+y^2+z^2)-(w+x+y+z)^2 =k \] for integer \(k\). The result can be applied to studying spherical and hyperbolic packings. For Part I, see {it R. L. Graham} et al., J. Number Theory 100, No. 1, 1--45 (2003; Zbl 1026.11058).