Question 71
Prove for any integer $n \ge 4$, there exists a polynomial of degree $n$,
$$
f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
$$
with the following properties.
(1) $a_0, a_1, \dots, a_{n-1}$ are all positive integers;
(2) For any positive integer $m$ and arbitrary $k$ ($k \ge 2$) positive integers $r_1, r_2, \dots, r_k$ that are different from each other, we have
$$
f(m) \neq f(r_1)f(r_2)\cdots f(r_k).
$$
$$
f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
$$
with the following properties.
(1) $a_0, a_1, \dots, a_{n-1}$ are all positive integers;
(2) For any positive integer $m$ and arbitrary $k$ ($k \ge 2$) positive integers $r_1, r_2, \dots, r_k$ that are different from each other, we have
$$
f(m) \neq f(r_1)f(r_2)\cdots f(r_k).
$$