Find all natural numbers $n$ satisfying $n+1 \mid n^{2}+1$
If $n+1 \mid n^{2}+1$, then $n+1 \mid (n+1)^{2}-(n^{2}+1)$, so $n+1 \mid 2n$, so $n+1 \mid 2n-2(n+1)$, so $n+1 \mid -2$, so $n=0$ or $n=1$.

Conversely, 0 and 1 work well.