Can any base ten repunit, other than 1, be a perfect cube?

If so, give an example. Otherwise prove that no base ten repunit (other than 1) can be a perfect cube.

Googling "repunit cube" results in most references saying that it is not known if there are any repunits that are cubes. A couple seem to disagree and quote a single source, and claim that there are no repunit perfect cubes. They don't show the proof, only point to, as I said, a single reference.

In any case, it would seem as difficult as solving the Riemann Hypothesis or Goldbach conjecture (at least a D5).

Posted by Charlie on 2011-10-25 12:38:22