12. Given that $n^{4}-3 n^{2}+9$ is a prime number, find all possible positive integer values of $n$.

 untranslated portion:
 已知 $n^{4}-3 n^{2}+9$ 是素数, 求正整数 $n$ 的所有可能值。

For a more accurate translation, the full text would be:

12. Given that $n^{4}-3 n^{2}+9$ is a prime number, find all possible positive integer values of $n$.

However, if you want to keep the original format and structure, here is the translation:

12. Given that $n^{4}-3 n^{2}+9$ is a prime number, find all possible positive integer values of $n$.

 untranslated portion:
 Given that $n^{4}-3 n^{2}+9$ is a prime number, find all possible positive integer values of $n$.
For $n=1$, $n^{4}-3 n^{2}+9=7$, and for $n=2$, $n^{4}-3 n^{2}+9=13$ are both prime numbers, satisfying the problem; for $n \geqslant 3$, $n^{4}-3 n^{2}+9=n^{4}+6 n^{2}+9-9 n^{2}=\left(n^{2}+3 n+3\right)\left(n^{2}-3 n+3\right)$, and $n^{2}-3 n+3 \geqslant 3$, thus $n^{4}-3 n^{2}+9$ is a composite number, not satisfying the problem.
Therefore, the only possible values of the positive integer $n$ are 1, 2.