Let $P\in \Bbb{Z}[X]$ be a polynomial with degree $d>1$ which is not divisible by $X$.

It is conjectured that for all such $P$, their range for integer inputs $R_P:=P(\Bbb{Z})$ has finite intersection with the set of factorials $\{n!:n\ge 0\}$. Here the assumption about divisibility by $X$ is not known to be needed.

I was curious if there are any counter-examples to the following stronger claim:

For all such $P$, $R_P$ does not contain an infinite sequence $a_1<a_2<\dots$ where $a_i \mid a_{i+1}$ for $i\ge 1$. Or even stronger, there exists a constant $C=C_P$ so that $R_P$ does not contain divisibility chains longer than $C$.

Let $P\in \Bbb{Z}[X]$ be a polynomial with degree $d>1$.

It is conjectured that for all such $P$, their range for integer inputs $R_P:=P(\Bbb{Z})$ has finite intersection with the set of factorials $\{n!:n\ge 0\}$.

We say that $P$ is “good” if there does not exist some $Q\in \Bbb{Z}[X]\setminus \{X\}$ such that $P \mid P\circ Q$. Examples: if $P=X^2$ then $Q=2X$ shows $P$ isn’t good; if $P=X^2-1$, then $Q=X^2$ shows $P$ isn’t good.

I was curious if there are any counter-examples to the following stronger claim:

For all such good $P$, $R_P$ does not contain an infinite sequence $a_1<a_2<\dots$ where $a_i \mid a_{i+1}$ for $i\ge 1$. Or even stronger, there exists a constant $C=C_P$ so that $R_P$ does not contain divisibility chains longer than $C$.

Also, is there a nice characterization for when $P$ is good?