Find all positive integers $n \le 6$, such that the equation $a^n+b^n=c^n+n$ has solutions over the integers. Find all positive integers $n \le 6$, such that the equation $a^n+b^n=c^n+n$ has solutions over the integers.
My attempt:
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 $n=4,5,6$?

If $n$ is such that $2n+1$ is prime, then the only $n$th powers mod $2n+1$ are $0$ and $\pm 1$ (since the group of units mod $2n+1$ is cyclic of order $2n$).  It follows easily that $a^n+b^n=c^n+n$ has no solutions mod $2n+1$ if $n>3$.  In particular, this handles the cases $n=5$ and $n=6$ (as well as infinitely many other larger values of $n$).
For $n=4$, you need a more ad hoc argument, since $2n+1$ is not prime.  Working mod $8$ does the trick, since the only $4$th powers (in fact, the only squares) mod $8$ are $0$ and $1$.
