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Apollonian circle packings: number theory. II: Spherical and hyperbolic packings. (English) Zbl 1165.11057

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).
Reviewer: Xu Fei (Beijing)

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11D09 Quadratic and bilinear Diophantine equations
52C26 Circle packings and discrete conformal geometry
11E12 Quadratic forms over global rings and fields

Citations:

Zbl 1026.11058
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References:

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