dc.contributor.author Northshield, Sam dc.date.accessioned 2018-04-09T18:04:27Z dc.date.accessioned 2020-06-22T14:35:31Z dc.date.available 2018-04-09T18:04:27Z dc.date.available 2020-06-22T14:35:31Z dc.date.issued 2006 dc.identifier.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 en_US dc.identifier.uri http://hdl.handle.net/20.500.12648/1121 dc.description This article has been published in the August 2006 issue of Journal of Number Theory. en_US dc.description.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. en_US dc.language en_US en_US dc.language.iso en_US en_US dc.publisher Journal of Number Theory en_US dc.subject Circle packings en_US dc.subject Apollonian circles en_US dc.subject Gaussian integers en_US dc.subject Totient en_US dc.title On integral Apollonian circle packings en_US dc.type Article en_US refterms.dateFOA 2020-06-22T14:35:31Z dc.description.institution SUNY Plattsburgh
﻿

Name:
fulltext.pdf
Size:
193.7Kb
Format:
PDF
Description:
full-text