Fibonacci-Factors Polynomial

Number Theory Level 5

Suppose $f$ is a polynomial with integer coefficients, such that for all positive integers $n$ the $n$ -th Fibonacci number $u_n$ divides $f(u_{n+1})$ . Find the smallest possible positive value of $f(4)$ .

Details and assumptions

The Fibonacci numbers are defined as follows:

$u_1=1,\ u_2=1;$ $u_n=u_{n-1}+u_{n-2}$ for all $n\geq 3$ .

0 is not a positive number.