MathDB

Let

f^{(n)}(x)

denote the

n^{\text{th}}

iterate of function

f

, i.e

f^{(1)}(x)=f(x)

f^{(n+1)}(x)=f(f^{(n)}(x))

. Let

p(n)

be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation

f^{(n)}(n)=p(n)

has exactly one solution

f

that maps the positive integers into the positive integers?
Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest

Problem Statement