1. Find all positive integers $m$ such that $m^{2}+m+7$ is a perfect square. ${ }^{[1]}$
Let $m^{2}+m+7=n^{2}$.
Multiplying both sides of the equation by 4 gives
$$
4 m^{2}+4 m+28=4 n^{2} \text {, }
$$

which is $4 n^{2}-(2 m+1)^{2}=27$.
Using the difference of squares formula, we solve to get $m=1,6$.