KöMaL 2022 A835

A. 835. Let $f^{(n)}(x)$ denote the $n^\text{th}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f\Big(f^{(n)}(x)\Big)$. 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?