For which positive integers $n$ is $n^{4}+4$ equal to a prime number?
For $n=1$ we get $1^{4}+4=5$, which works.

For all other values of $n$, the key idea is that

$$
n^{4}+4=n^{4}+4 n^{2}+4-4 n^{2}=\left(n^{2}+2\right)^{2}-(2 n)^{2}=\left(n^{2}+2 n+2\right)\left(n^{2}-2 n+2\right)
$$

which is the product of two integers greater than 1, and hence cannot be prime.