What about other residue class restrictions besides mod 9 ? Googling, I found this discussion, <https://mathoverflow.net/questions/138886/which-integers-can-be-expressed-as-a-sum-of-three-cubes-in-infinitely-many-ways>, where someone states no other such restrictions are known. Most interesting of all, it is apparently unsolved whether for *every* N ≠ ±4 (mod 9) the equation x^3 + y^3 + z^3 = N has infinitely many integer solutions in x, y, z. This is known for N = a perfect cube or twice a perfect cube, but it seems in no other case. Another cute tidbit: N = 3 has two known solutions mod permutations: 1 + 1 + 1 = 3 and 4^3 + 4^3 + (-5)^3 = 3. Are there any more? Not known. —Dan Allan Wechsler wrote: ----- Aside from n = 4 or 5 mod 9, which are all impossible, the next unresolved case is x^3 + y^3 + z^3 = 42. -----