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.


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