Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 + y2 + z2 + w2 = 1/2(x + y + z + w 2. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element -n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple. © 2003 Elsevier Science (USA). All rights reserved.
Graham, R. L., Lagarias, J. C., Mallows, C. L., Wilks, A. R., & Yan, C. H. (2003). Apollonian circle packings: Number theory. Journal of Number Theory, 100(1), 1–45. https://doi.org/10.1016/S0022-314X(03)00015-5