On integral Apollonian circle packings

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.