Article ID Journal Published Year Pages File Type
4595628 Journal of Number Theory 2006 23 Pages PDF
Abstract

The curvatures of four mutually tangent circles with disjoint interiors form what is called a Descartes quadruple. The four least curvatures in an integral Apollonian circle packing form what is called a root Descartes quadruple and, if the curvatures are relatively prime, we say that it is a primitive root quadruple. We prove a conjecture of Mallows by giving a closed formula for the number of primitive root quadruples with minimum curvature −n. An Apollonian circle packing is called strongly integral if every circle has curvature times center a Gaussian integer. The set of all such circle packings for which the curvature plus curvature times center is congruent to 1 modulo 2 is called the “standard supergasket.” Those centers in the unit square are in one-to-one correspondence with the primitive root quadruples and exhibit certain symmetries first conjectured by Mallows. We prove these symmetries; in particular, the centers are symmetric around y=x if n is odd, around x=1/2 if n is an odd multiple of 2, and around y=1/2 if n is a multiple of 4.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory