Assume that both $ k+1$ and $ n^{k} +k$ are prime number for some $ ( n,k) \in \mathbb{Z}^{2}$ with $ 2\leq n\leq k$. Since $ k+1$ is a prime and $ \gcd( n,k+1) =1$, $ n^{k} +k\equiv n^{k} -1\equiv 0\pmod{k+1}$, which is impossible. Thus $ n$ must be $ 1$, and hence $ n+k=k+1$ is a prime number, of course. $\quad$ $ \blacksquare $