For which positive integers $n$, does the number $n^{2}+1$ divide $n+1$?
If $n^{2}+1$ divides $n+1$, we must have $n^{2}+1 \leq n+1$, or equivalently $n^{2} \leq n$, that is, $n \leq 1$. The only possibility is therefore $n=1$. Conversely, $n=1$ does work.