For which natural numbers $n$ is the integer $n^{2}+n+1$ a perfect square?
If $n=0$, we have $n^{2}+n+1=1$ which is a perfect square. Otherwise, we have

$$
n^{2}<n^{2}+n+1<n^{2}+2 n+1=(n+1)^{2}
$$

thus $n^{2}+n+1$ is strictly between two consecutive squares and cannot be one.