Find all positive integers $n$ such that $2^{n}+12^{n}+2011^{n}$ is a perfect square.
The answer $n=1$ works, because $2^{1}+12^{1}+2011^{1}=45^{2}$. We prove it's the only one.
- If $n \geq 2$ is even, then modulo 3 we have $2^{n}+12^{n}+2011^{n} \equiv 1+0+1 \equiv 2(\bmod 3)$ so it is not a square.
- If $n \geq 3$ is odd, then modulo 4 we have $2^{n}+12^{n}+2011^{n} \equiv 0+0+3 \equiv 3(\bmod 4)$ so it is not a square.

This completes the proof.