Some Experiments with Integral Apollonian Circle Packings

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 [ Sarnak 07 ], 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.