[seqfan] Re: Integral Apollonian circle packings

[Neil Sloane]

I asked Jeff Lagarias Rick's question, The answer is No. Jeff said:
(1) The answer is: No,
repeated equal curvatures exist in any
integer Apollonian packing.

 (2) Reason. The number of circles in an Apollonian packing
(integral or not) of curvature at most r  is always:

 >>  r^{hausdorff dimension} > r^{1.30}

as r \to \infty. See Theorem 5.2 of journal number theory paper
[GLMWY] (result due to david boyd)
To have uniqueness would need this number
to be at most r.

Best regards
Neil



On Sun, Nov 8, 2015 at 6:09 PM, Rick Shepherd <rlshepherd2 at gmail.com> wrote:

> Do integral Apollonian circle packings  exist where the curvature of each
> circle in the packing is distinct? If so, of particular interest would be
> the "least" such sequence of curvatures.
> (Is there such a packing of the unit circle?)
>
> Thanks,
> Rick
>
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