Some experiments with integral Apollonian circle packings

TL;DR

This paper investigates prime curvatures in integral Apollonian circle packings, providing heuristic estimates, numerical data, and experimental evidence for the distribution of prime circles and kissing prime pairs.

Contribution

It offers the first heuristic and numerical analysis of prime curvatures and kissing prime pairs in primitive integral Apollonian circle packings, supported by experimental data.

Findings

Heuristic estimates for the count of prime curvatures less than x.

Numerical data supporting the distribution of prime circles.

Evidence suggesting a local to global principle for curvatures.

Abstract

Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a 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 \cite{ll}, 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. 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 {\it pairs} of circles of prime curvature less than $x$ in a primitive integral ACP. We also provide experimental evidence towards a local to global principle for the curvatures in a primitive integral ACPs.