Naslov
Strong Density Conjecture for Integral Apollonian Circle Packing

Autori
Martinjak, Ivica

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Four Faces of Number Theory / - , 2012

Skup
Four Faces of Number Theory

Mjesto i datum
Würzburg, Njemačka, 07.08.2012. - 12.08.2012

Vrsta sudjelovanja
Poster

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Apollonian circle packing ; strong density conjecture ; twin prime conjecture

Sažetak
Given three mutually tangent circles it is always possible to find two circles tangent to all three. Starting with a such triple of circles one can create a set with one outer circle called Apollonian circle packing (ACP). While there are many papers on geometric aspects of Apollonian packing and its generalizations, questions on the diophantine properties are quite recent. P. Sarnak proved that in a primitive integral ACP there are infinitely many circles with prime curvature. Moreover, in this set an analogy with twin prime conjecture holds: there are infinitely many pairs of tangent circles with a prime curvature. Here we deal with the strong density conjecture for the packing. The conjecture claims that for any ACP there exist $X \in {; ; \mathbb Z}; ; $ such that any $x > X$ whose residue mod 24 lies in the set of residue classes mod 24 appears as a curvature. Based on a few instances of ACP our numerical results contribute to the strong density conjecture for curvatures in an integral Apollonian packing. The frequencies with which integers of the given range satisfying the congruence condition appear as a curvatures tend to normal distribution having as less exceptions as the range increases.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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