The specific question is find all solutions for
$(x+y)^{2012} = x^{2012} + y^{2012}$ but obviously, the number isn't "important."
It would be better to find a general solution for
$(x+y)^{n} = x^{n} + y^{n}$
It is immediately apparent that one possible solution is $(0,0)$ and taking this further
$(0,m)$ and $(m,0)$ where $m$ is an integer when $n$ is even.
However, what about the cases where both $(x,y)$ are not zero?
Possibly $(i,j) \neq 0$
Is this even possible?