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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,”
http://publications.ias.edu/sites/default/files/AppolonianPackings.pdf]
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.

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
52C26 Circle packings and discrete conformal geometry
11Y60 Evaluation of number-theoretic constants
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