Oh, for some reason, I thought that once you had one solution there was known technology for finding an infinite sequence of them. On Sat, Mar 9, 2019 at 3:45 PM Dan Asimov <dasimov@earthlink.net> wrote:
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. -----
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