Some experiments with integral Apollonian circle packings. (English) Zbl 1259.11065

Summary: Bounded Apollonian circle packings (ACPs) are constructed by repeatedly inscribing circles into the triangular interstices of a Descartes configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In [“Letter to Lagarias about integral Apollonian packings June, 2007,”
P. Sarnak proves that there are infinitely many circles of prime curvature and infinitely many pairs of tangent circles of prime curvature in a primitive integral ACP. (A primitive integral ACP is one in which no integer greater than 1 divides the curvatures of all of the circles in the packing.) In this paper, we give a heuristic backed up by numerical data for the number of circles of prime curvature less than \(x\) and the number of “kissing primes”, or pairs of circles of prime curvature less than \(x\), in a primitive integral ACP. We also provide experimental evidence toward a local-to-global principle for the curvatures in a primitive integral ACP.


11H31 Lattice packing and covering (number-theoretic aspects)
52C26 Circle packings and discrete conformal geometry
11Y60 Evaluation of number-theoretic constants
Full Text: DOI arXiv Euclid


[1] DOI: 10.1090/S0894-0347-2011-00707-8 · Zbl 1228.11035
[2] DOI: 10.1007/s00222-009-0225-3 · Zbl 1239.11103
[3] Cassels [Cassels 78] J. W. S., Rational Quadratic Forms (1978) · Zbl 0395.10029
[4] Cogdell [Cogdell 03] J., J. Théor. Nombres Bordeaux 15 pp 34– (2003)
[5] Coxeter [Coxeter 05] H. S. M., Non-Euclidean Geometries, János Bolyai Memorial Volume pp 109– (2005)
[6] DOI: 10.1007/BF01234411 · Zbl 0692.10020
[7] Fuchs [Fuchs 10] E., ”Arithmetic Properties of Apollonian Circle Packings.” (2010)
[8] DOI: 10.1016/j.jnt.2011.05.010 · Zbl 1259.11066
[9] DOI: 10.1016/S0022-314X(03)00015-5 · Zbl 1026.11058
[10] DOI: 10.1073/pnas.29.11.378 · Zbl 0063.03162
[11] DOI: 10.1090/S0894-0347-2011-00691-7 · Zbl 1235.22015
[12] Sanden [Sanden 09] K., ”Integral Apollonian Circle Packings: A Prime Number Conjecture.” (2009)
[13] Sarnak, [Sarnak 07] P. 2007. ”Letter to Lagarias.” Available online (http://www.math.princeton.edu/sarnak).
[14] DOI: 10.4169/amer.math.monthly.118.04.291 · Zbl 1260.52011
[15] Sedgewick, [Sedgewick and Wayne 07] R. and Wayne, K. 2007. ”Introduction to Programming in Java.” Available online (http://www.cs.princeton.edu/introcs/home).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.