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deg P = 1 if sequence contains a b−th power of some positive integer >1

Source: 2021 Saudi Arabia Training Lists p35 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022

polynomialInteger Polynomialalgebra

Problem Statement

Let

P (x)

be a non constant integer polynomial and positive integer

n

. The sequence

a_0, a_1, ...

is defined by

a_0 = n

and

a_k = P (a_{k-1})

for

k \ge 1

. Given that for each positive integer

b

, the sequence contains a

b

-th power of some positive integer greater than

1

. Prove that deg

P = 1