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

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This is a current problem at brilliant.org: brilliant.org/i/C4F9a6. Moderators, please note. –  Jon Haussmann Dec 29 '12 at 0:49
The above question is posted as a Challenge problem on Brilliant.org, which offers weekly problem sets to test student's problem solving abilities. John Chang has been posting questions on math.stackexchange.com and expecting others to solve the problems for him. He has posted another one of our questions at math.stackexchange.com/questions/238677/… -Calvin Lin Mathematics Challenge Master –  Calvin Lin Dec 29 '12 at 1:05

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This is apparently a current problem at brilliant.org (h/t John Haussmann). I'm taking out my answer for now.

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