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Another Sequence with P^n(x)

Source: Swiss TST 2019 P4

May 12, 2020
number theorypolynomialprime numbersalgebra

Problem Statement

Let pp be a prime number. Find all polynomials PP with integer coefficients with the following properties: (a)(a) P(x)>xP(x)>x for all positive integers xx. (b)(b) The sequence defined by p0:=pp_0:=p, pn+1:=P(pn)p_{n+1}:=P(p_n) for all positive integers nn, satisfies the property that for all positive integers mm there exists some l≥0l\geq 0 such that m∣plm\mid p_l.
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