Find all integers $n \in \mathbb{N}$ such that $(n+1) \mid\left(n^{2}+1\right)$.
Suppose $n$ is a solution. Since $(n+1) \mid (n+1)^{2}$, then $(n+1) \mid \left((n+1)^{2} - (n^{2} + 1)\right)$, i.e., $(n+1) \mid 2n$, and then $(n+1) \mid (2(n+1) - 2n)$, i.e., $(n+1) \mid 2$, which means $n \in \{0, 1\}$.

Conversely, we verify that 0 and 1 work.

Thus, the solutions are 0 and 1.