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I was wondering if it is possible to construct an Apollonian gasket where every circle has a unique integer curvature.

Take for instance the following gasket, defined by curvatures (−10, 18, 23, 27):

Apollonian gasket

At first glance this looks like a candidate, but for a curvature of 242 there are already three circles that we can find.

Now, knowing that there are infinitely many circles in a gasket, I would guess that any gasket will at some point have two or more circles with the same curvature. But can this be proven?

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Take a look at the values you're getting, mod 12.

All the values are $2, 3, 6, 11$ (mod 12).

If you look at more of these gaskets, you'll find that they always can only hit 4 values, mod 12. That severely limits the possible spread of values. From there, you can use the pigeonhole principle on the possible values and the number of circles that need to be filled.

My Circles of Descartes demo has code for playing around with these.

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