15. Let $f(x)$ be a polynomial with integer coefficients, and let $P=\left\{p \mid p\right.$ be a prime and for some $\left.j \in \mathbf{N}^{*}, p \mid f\left(2023^{j}\right)\right\}$. Given that $P$ is a finite set, find $f(x)$.
Solve
$f(x)=c x^{n}$, where $c$ is a non-zero integer.
Prove by induction on the degree of $f$ that it suffices to show that the constant term of $f$ is zero when the degree of $f$ is non-zero. At this time, $f(x)=x g(x)$, then $g(x)$ satisfies the same condition.
Let $P=\left\{p_{1}, p_{2}, \cdots, p_{r}\right\}$, and $f(2023)=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{r}^{\alpha_{r}}, \alpha_{i} \geqslant 0, i=1,2, \cdots, r$.
Denote $m=\varphi\left(\prod_{p_{j} \neq 7,17} p_{j}^{\alpha_{j}+1}\right)$, let $n-1=k m$, where $k \in \mathbf{N}^{*}$,
then for all prime numbers $p_{j} \neq 7,17$, $2023^{n}-2023=2023\left(2023^{n-1}-1\right) \equiv 0\left(\bmod p_{j}^{\alpha_{j}+1}\right)$. By $\left(2023^{n}-2023\right) \mid\left(f\left(2023^{n}\right)-f(2023)\right)$ we know that when $n-1=k m$,
$$
f\left(2023^{n}\right)=\prod_{p_{j} \neq 7,17} p_{j}^{\alpha_{j}} \cdot 7^{\beta_{k}} \cdot 17^{\gamma_{k}} \text {. }
$$

If the constant term $c$ of $f$ is non-zero, then when $k$ is sufficiently large,
$$
v_{7}(c), v_{17}(c)<1+k m \text {, thus } \beta_{k}=v_{7}\left(f\left(2023^{n}\right)\right)=v_{7}(c), \gamma_{k}=v_{17}\left(f\left(2023^{n}\right)\right)=v_{17}(c) \text {. }
$$

Therefore, $f\left(2023^{n}\right)$ is a constant, which contradicts the non-zero degree of $f$.