Problem: Find all positive integers $n$ such that $2^n+12^n+2011^n$ is a perfect square.

Solution: The answer $n=1$ works, because $2^1+12^1+2011^1=45^2$.
We prove it's the only one.
\begin{itemize}
  \ii If $n \ge 2$ is even, then modulo $3$ we have
  $2^n+12^n+2011^n \equiv 1+0+1 \equiv 2 \pmod 3$
  so it is not a square.

  \ii If $n \ge 3$ is odd, then modulo $4$ we have
  $2^n+12^n+2011^n \equiv 0+0+3 \equiv 3 \pmod 4$
  so it is not a square.
\end{itemize}
This completes the proof.