Find all integers $n$ such that $4 n^{4}+1$ is a prime number.
By the Sophie-Germain identity:

$1+4 n^{4}=\left(1-2 n+2 n^{2}\right)\left(1+2 n+2 n^{2}\right)$ Therefore, it is necessary that $1-2 n+2 n^{2}=1$, so $n=1$.