Since giving the recent talk, I had a bit of a breakthrough. Two of the problems I mentioned are no longer open! I now know (and can prove) the following:
Theorem: If are nonzero integers, at most one of which is negative, then the number of inequivalent occurrences of three mutually tangent spheres of curvature is the same as the number of algebraic integers in with norm , up to multiplication by a unit. The bijection between algebraic integers with that norm and triples of spheres is natural and explicit.
There is a nice way to compute this number. Define an arithmetic function on the positive integers as follows. If or , then if is even and if is odd; if , then ; for all . Extend to all positive integers by multiplicativity. Then .
Theorem: If are nonzero integers, at most one of which is negative, then the number of inequivalent occurrences of tangent spheres of curvature is given by the formula . Here is the number of solutions to the congruence , is the number of solutions to the congruence , and is 1 if and , otherwise 0.
(This strange formula comes from an application of Burnside’s Counting Lemma; as always in these problems, the trickiest part is keeping track of which solutions correspond to the same packing.)
The next natural problem in the progression, counting the total number of inequivalent occurrences of a given curvature in integer sphere packings, remains resistant to my current approach.
In the next week, I should have the corresponding result for counting occurrences of a given -tuple of -dimensional hyperspheres. If I’m lucky, my 2- and 3-dimensional techniques will generalize to let me count occurrences of -tuples in dimensions, but I can’t be sure of that part yet. Beyond that, I currently have no idea how to proceed.