Find the set of positive integers $n$ such that $5^{n}+4$ is a perfect square

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Let $a>0$ such that $5^{n}+4=a^{2}$, which gives us $5^{n}=(a-2)(a+2)$. Therefore, the two terms on the right must be powers of 5. However, they are not congruent modulo 5, so one of them must be a zero power. Thus, $a-2=1 \Rightarrow a=3$. We then have $5^{n}+4=9 \Rightarrow n=1$. The only solution is therefore $n=1$.