GS. 1 Given that $n$ is a positive integer. If $n^{2}+5 n+13$ is a perfect square, find the value of $n$.
$\begin{array}{l}n^{2}+5 n+13=n^{2}+5 n+2.5^{2}-2.5^{2}+13=(n+2.5)^{2}+6.75=m^{2} \text {, where } m \text { is an integer } \\ m^{2}-(n+2.5)^{2}=6.75 \Rightarrow(m+n+2.5)(m-n-2.5)=6.75 \Rightarrow(2 m+2 n+5)(2 n-2 m-5)=27 \\ \left\{\begin{array}{l}2 m+2 n+5=27 \\ 2 m-2 n-5=1\end{array} \text { or }\left\{\begin{array}{l}2 m+2 n+5=9 \\ 2 m-2 n-5=3\end{array}\right.\right. \\ n=4 \text { or } n=-1 \text { (rejected, } \because n>0 \text { ) }\end{array}$