Gap Principle of Divisibility Sequences of Polynomials

Let $f \in \mathbb{Z}[x]$ and $\ell \in \mathbb{N}$. Consider the set of all $(a_{0},a_{1},\dots,a_{\ell}) \in \mathbb{N}^{\ell+1}$ with $a_{i} < a_{i+1}$ and $f(a_{i}) \mid f(a_{i+1})$ for all $0 \leq i \leq \ell-1$. We say that $f$ satisfies the gap principle of order $\ell$ if $\lim a_{\ell}/a_{0} = \infty$ as $a_{0} \to \infty$ for any such $(a_{0},a_{1},\dots,a_{\ell})$. We also define the gap order of $f(x)$ to be the smallest positive integer $\ell$ such that $f(x)$ satisfies the gap principle of order $\ell$. If such $\ell$ does not exist, we say that $f(x)$ does not satisfy the gap principle. In this talk, we will discuss a conjecture by Chan, Choi and Lam that $f(x)$ does not satisfy the gap principle  if and only if $f(x)$ is in the form of $f(x)=A(Bx+C)^n$ for some $A, B, C \in \mathbb{Z}$. Moreover, we completely determine the gap order of any polynomial  that if $f(x)$ is not in the form of $A(Bx+C)^n$, then $f(x)$ has gap order $2$ if $f(x)$ is a quadratic polynomial or a power of a quadratic polynomial; and has gap order $1$ otherwise.  Related to the proof of above results, the multiplicative order of the fundamental solution of Pell's equation $X^2-DY^2=1$ in $\mathbb{Z} [\sqrt{D}]/<D>$ will also be discussed. These are joint work with Tsz Ho Chan, Peter Cho-Ho Lam and Daniel Tarnu.