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dc.contributor.authorNorthshield, Sam
dc.date.accessioned2018-04-09T18:04:27Z
dc.date.accessioned2020-06-22T14:35:31Z
dc.date.available2018-04-09T18:04:27Z
dc.date.available2020-06-22T14:35:31Z
dc.date.issued2006
dc.identifier.citationNorthshield, S. (2006). On integral Apollonian circle packings. Journal of Number Theory, 119(2). http://doi.org/10.1016/j.jnt.2005.10.003en_US
dc.identifier.urihttp://hdl.handle.net/20.500.12648/1121
dc.descriptionThis article has been published in the August 2006 issue of Journal of Number Theory.en_US
dc.description.abstractThe 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.en_US
dc.languageen_USen_US
dc.language.isoen_USen_US
dc.publisherJournal of Number Theoryen_US
dc.subjectCircle packingsen_US
dc.subjectApollonian circlesen_US
dc.subjectGaussian integersen_US
dc.subjectTotienten_US
dc.titleOn integral Apollonian circle packingsen_US
dc.typeArticleen_US
refterms.dateFOA2020-06-22T14:35:31Z
dc.description.institutionSUNY Plattsburgh


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