THE 68th NMO SELECTION TESTS FOR THE JUNIOR BALKAN MATHEMATICAL OLYMPIAD

For $d=1$ it follows that $1+k$ is a prime.

For $d=n$ it follows that $n^k+k$ is a prime. As $k+1$ does not divide $n$ (being larger than $n$), from Fermat's Theorem we get $n^k\equiv 1(\mathrm{mod}k+1)$, hence $n^k+k\equiv 0(\mathrm{mod}k+1)$. It follows that $n^k+k=k+1$, hence $n=1$. We have seen at the beginning that $1+k$ is a prime.