Question 137
For any positive integer $k$, let $S(k)$ denote the sum of its digits in base ten (e.g. $S(209) = 2 + 0 + 9 = 11$). Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n \ge 2017$ we have $P(n) \gt 0$ and
$$
S(P(n)) = P(S(n)).
$$
$$
S(P(n)) = P(S(n)).
$$