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35

Part of 2021 Saudi Arabia Training Tests

Problems(1)

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

1/5/2022
Let P(x)P (x) be a non constant integer polynomial and positive integer nn. The sequence a0,a1,...a_0, a_1, ... is defined by a0=na_0 = n and ak=P(ak−1)a_k = P (a_{k-1}) for k≥1k \ge 1. Given that for each positive integer bb, the sequence contains a bb-th power of some positive integer greater than 11. Prove that deg P=1P = 1
polynomialInteger Polynomialalgebra