Question 159
Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant?
(A *Fibonacci number* is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0 = 0$, $F_1 = 1$, and $F_{k+2} = F_{k+1} + F_k$ for $k \ge 0$.)
(A *Fibonacci number* is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0 = 0$, $F_1 = 1$, and $F_{k+2} = F_{k+1} + F_k$ for $k \ge 0$.)