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):
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?