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    On integral Apollonian circle packings

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    Author
    Northshield, Sam
    Keyword
    Circle packings
    Apollonian circles
    Gaussian integers
    Totient
    Date Published
    2006
    
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    URI
    http://hdl.handle.net/20.500.12648/1121
    Abstract
    The curvatures of four mutually tangent circles with disjoint interior form what is called a Descartes quadruple. The four smallest curvatures of circles in an 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 center of the largest circle is in the unit square and for which curvature plus curvature times center is congruent to 1 modulo 2 is called the standard super-gasket. These centers are in oneto-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.
    Citation
    Northshield, S. (2006). On integral Apollonian circle packings. Journal of Number Theory, 119(2). http://doi.org/10.1016/j.jnt.2005.10.003
    Description
    This article has been published in the August 2006 issue of Journal of Number Theory.
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