Find all prime numbers $p$ such that $5^p+ 4p^4$ is a perfect square

Question

Find all prime numbers $p$ such that $5^p+ 4p^4$ is a perfect square.

Answer 1

$5$ is the only such prime. Note that if $$5^p+4p^4=n^2$$ Then $$5^p=(n+2p^2)(n-2p^2)$$ Each factor must be a power of $5$, hence the difference of the factors is a multiple of $5$ unless $5^p=4p^2+1$. If this is not the case, then this difference is $4p^2$ and a multiple of $5$, so $p$ is a multiple of $5$. Since it is prime, $p=5$.

Otherwise, since $p^2>1$ we have $5p^2>5^p$, so $p^2>5^{p-1}$. By induction we can prove that this entails that $p<5$, so this is impossible to satisfy given the conditions.