Find all positive integers $n \le 6$, such that the equation $a^n+b^n=c^n+n$ has solutions over the integers.
For $n=1$, trivially there are integer solutions.
For $n=2$, we have the solutions $$a=b=\pm 1, c=0.$$
For $n=3$, we have the solution $$a=b=1, c = -1.$$
Anyone can complete the proof for $n=4,5,6$?