Example 13 (25th Brazilian Mathematical Olympiad) Find the smallest positive prime such that for some integer $n$, this prime divides $n^{2}+5 n+23$.

Will the translated text be used for further discussion or do you need the solution to this problem as well?
Let $f(n)=n^{2}+5 n+23$. Since
$$
\begin{array}{l}
f(n) \equiv 1(\bmod 2), f(n) \equiv \pm 1(\bmod 3), \\
f(n) \equiv-1, \pm 2(\bmod 5), \\
f(n) \equiv 1,3, \pm 2(\bmod 7), \\
f(n) \equiv 1, \pm 3, \pm 4,6(\bmod 11), \\
f(n) \equiv-2, \pm 3,4,-5, \pm 6(\bmod 13),
\end{array}
$$

and $f(-2)=17$,
the smallest positive prime number that satisfies the condition is 17.