A long time ago I found a question on the internet that went a little like this:
Suppose that we have $n=2^k$ where $k\gt 3$. If $m$ is another number that is a combination of the digits of $2^k$, prove that $m$ cannot be a power of $2$.
I gave up on it a long time ago, but have now become interested in number theory and hope that someone could shed some light on this problem.
Edit: This is only for base 10.