$S$-parts of terms of integer linear recurrence sequences

Abstract

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively prime to $q_1 \ldots q_s$. We define the $S$-part $[m]_S$ of $m$ by $[m]_S := q_1^{r_1} \ldots q_s^{r_s}$. Let $(u_n)_{n \ge 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\varepsilon > 0$, there exists an integer $n_0$ such that $[u_n]_S\leq |u_n|^{\varepsilon}$ holds for $n > n_0$. Our proof is ineffective in the sense that it does not give an explicit value for $n_0$. Under various assumptions on $(u_n)_{n \ge 0}$, we also give effective, but weaker, upper bounds for $[u_n]_S$ of the form $|u_n|^{1 -c}$, where $c$ is positive and depends only on $(u_n)_{n \ge 0}$ and $S$.