Some experiments with integral Apollonian circle packings

Details Some experiments with integral Apollonian circle packings Some experiments with integral Apollonian circle packings

2010-01-11

arxiv.org/abs/1001.1406v1

Mathematics

Number Theory

30 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Some experiments with integral Apollonian circle packings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Some experiments with integral Apollonian circle packings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Some experiments with integral Apollonian circle packings will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-458324