## Experimental Mathematics

- Experiment. Math.
- Volume 20, Issue 4 (2011), 380-399.

### Some Experiments with Integral Apollonian Circle Packings

Elena Fuchs and Katherine Sanden

#### 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.

#### Article information

**Source**

Experiment. Math., Volume 20, Issue 4 (2011), 380-399.

**Dates**

First available in Project Euclid: 8 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1323367153

**Mathematical Reviews number (MathSciNet)**

MR2859897

**Zentralblatt MATH identifier**

1259.11065

**Subjects**

Primary: 11Y60: Evaluation of constants

**Keywords**

Number theory computational number theory Diophantine equations

#### Citation

Fuchs, Elena; Sanden, Katherine. Some Experiments with Integral Apollonian Circle Packings. Experiment. Math. 20 (2011), no. 4, 380--399. https://projecteuclid.org/euclid.em/1323367153