Question 141
Given a polynomial $P(x)$ with integer coefficients, assume that for every positive integer $n$ we have $P(n) \gt n$. Consider the sequence
$$
x_{1} = 1,\quad x_{2} = P(x_{1}),\quad \ldots,\quad x_{n} = P(x_{n-1}),\quad \ldots
$$
If for every positive integer $N$ there exists a member of the sequence divisible by $N$, prove that $P(x) = x + 1$.
$$
x_{1} = 1,\quad x_{2} = P(x_{1}),\quad \ldots,\quad x_{n} = P(x_{n-1}),\quad \ldots
$$
If for every positive integer $N$ there exists a member of the sequence divisible by $N$, prove that $P(x) = x + 1$.