first solution: by Zsigmondy theorem, $ x^n + y^n$ has at least 2 different prime factors (a prime factor of $ x + y$, and a primitive prime divisor of $ x^n + y^n$) - a contradiction, so the problem is trivial.
second solution: by the identity $ (a + p^kl)^{p^s t} - a^{p^s t} \equiv alt p^{k + s} \pmod {p^{k + s + 1}}$ for $ k > 0, s \ge 0$ (can be proved by induction on $ s$), we deduce the "Lifting the Exponent" lemma: $ ord_p{x^n + y^n} = ord_p{(x + y)} + ord_p{n}$ (for odd n, when x+y is divisibly by p). x+y must be divisible by p in our case (because $ x + y | x^n + y^n = p^k$ ), so we conclude: $ k = ord_p{(x + y)} + ord_p{n}$. Let $ m = ord_p{n}$. We have, by the same lemma, $ p^k | x^{p^m} + y^{p^m}$. We must have $ n = p^m$, else:
$ x^n + y^n > x^{p^m} + y^{p^m} \ge p^k$, a contradiction.
[Can someone please post a proof of Zsigmondy's Theorem?]