Summer Haag, Clyde Kertzer, James Rickards, Katherine E. Stange
Published 2023-07-06Version 1
In a primitive integral Apollonian circle packing, curvatures that appear must fall into one of six or eight residue classes modulo 24. The Local-Global Conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. We then formulate a new conjecture, and give computational evidence in support of it.
Comments: 25 pages, 4 figures
Products in Residue Classes
On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$
On the values of Dedekind sums
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