I should think you could solve a 4x4x4 cube by applying 3x3x3 moves, using
different combinations for the "center slice" - ie, rotating the center
two slices together allows all the regular corner moves; it also allows
switching adjacent edge pieces (in pairs) by using whatever you use to flip
edges in 3x3x3. Using only one of the center slices as "center", and
turning the other with a face should allow flipping a pair of L-R pieces,
so that L becomes R and vice versa. The center 4 pieces (which I haven't
thought about carefully) probably can be changed around at
(even parity) will, sometimes treating them as edges (since they can be
carried along on certain of the "edge" moves), and sometimes as centers
(for instance, to rotate a group of 4 of them halfway around in place).
I don't know what new parity limitations exist; nor do I know if
the same type of sequences are efficient for solving (ie, top-middle-bottom,
etc), but I shouldn't think the new cubes will be so very much more
I would assume the 5x5x5's could be handled similarly, except, of course,
they have a real center.