All this talk of Gaussian integers reminded me of a problem that I'd dearly love to see solved. Consider N bodies, each with integer masses. Start them in initial positions in a 2D (for the moment, there's plenty of complexity in 2D) which are on integer coordinates. The attractive Newtonian gravitation force is proportional to m*m'/(dx^2+dy^2), so this force is rational. We need to compute new positions for each of the bodies in such a way that all the conservation laws are preserved exactly (if this is possible), yet the number of bits required doesn't grow too quickly (see below). We need to preserve the total energy, we need to preserve momentum & angular momentum, & we might like to preserve information (the system is reversible). Note that we don't necessarily require that delta-t be constant, although if it isn't, we may need to have another counter to represent the "clock" -- i.e., the global time. I don't know how to do this even in the 2-body case, but the Minsky circle hack may provide some guidance -- if we could get the two bodies to each draw a Minsky circle while revolving around one another, then we'd have a good start on the problem. In addition to solving the problem of finite arithmetic, we also need to figure out what happens in collisions. Taking a cue from Fredkin, we should probably have 100% elastic collisions in order to ensure reversibility. How does this square with Poincaré ? Well, I'm assuming that my N-body problem for finite N is Turing complete, so with enough bodies and enough "tape" (empty space), we should be able to compute any computable function. Minsky already provides one potential representation with his "counter machines". So we want a reasonably "efficient" Turing Machine (or Counter Machine) simulation so that we aren't wasting more than a fixed fraction of the bits required to represent the state (N pairs of integers or N gaussian integers). Has anyone attempted this sort of exact simulation before?