TY - JOUR

T1 - Some experiments with integral apollonian circle packings

AU - Fuchs, Elena

AU - Sanden, Katherine

N1 - Funding Information:
We thank Peter Sarnak, Alex Kontorovich, and Kevin Wayne for many insightful conversations and helpful suggestions. We also thank the referee for valuable comments and remarks. K. Sanden was supported by NSF Grant 0758299.

PY - 2011/11/28

Y1 - 2011/11/28

N2 - 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.

AB - 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.

KW - Diophantine equations

KW - Number theory

KW - computational number theory

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U2 - 10.1080/10586458.2011.565255

DO - 10.1080/10586458.2011.565255

M3 - Article

AN - SCOPUS:84857995415

SN - 1058-6458

VL - 20

SP - 380

EP - 399

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 4

ER -