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2011 Some Experiments with Integral Apollonian Circle Packings
Elena Fuchs, Katherine Sanden
Experiment. Math. 20(4): 380-399 (2011).

Abstract

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," 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.

Citation

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Elena Fuchs. Katherine Sanden. "Some Experiments with Integral Apollonian Circle Packings." Experiment. Math. 20 (4) 380 - 399, 2011.

Information

Published: 2011
First available in Project Euclid: 8 December 2011

zbMATH: 1259.11065
MathSciNet: MR2859897

Subjects:
Primary: 11Y60

Keywords: computational number theory , Diophantine equations , number theory

Rights: Copyright © 2011 A K Peters, Ltd.

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Vol.20 • No. 4 • 2011
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