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

TL;DR

This paper investigates the prime factorization structure of terms in integer linear recurrence sequences, establishing bounds on the product of specified primes dividing these terms, under certain conditions.

Contribution

It proves that, under certain conditions, the S-part of recurrence sequence terms grows slower than any positive power of the term's absolute value, and provides effective bounds in some cases.

Findings

For large n, [u_n]_S ≤ |u_n|^ε for any ε > 0.

Effective bounds of the form |u_n|^{1 - c} are established under additional assumptions.

The results apply to the prime factorization properties of linear recurrence sequence terms.

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 $S$.