Problem

Source: Tuymaada 2003, day 2, problem 4.

Tags: algebra, polynomial, calculus, integration, number theory proposed, number theory



Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$ Proposed by F. Petrov

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