Fibonacci-Factors Polynomial

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

Details and assumptions

The Fibonacci numbers are defined as follows:

u1=1, u2=1;u_1=1,\ u_2=1; un=un1+un2u_n=u_{n-1}+u_{n-2} for all n3n\geq 3.

0 is not a positive number.


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