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Polynomial gcd

Source: Nigeria senior mathematics Olympiad round 2 problem 4.

September 8, 2019

algebranumber theorygreatest common divisor

Problem Statement

Let

h(t)

and

f(t)

be polynomials such that

h(t)=t^2

and

f_n(t)=h(h(h(h(h...h(t))))))-1

where

h(t)

occurs

n

times. Prove that

f_n(t)

is a factor of

f_N(t)

whenever

n

is a factor of

N