$S$-PARTS OF TERMS OF INTEGER LINEAR RECURRENCE SEQUENCES. Issue 3 (29th November 2017) - Bugeaud, Yann; Evertse, Jan-Hendrik
Abstract : Let $S=\{q_{1}, …, q_{s}\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m=q_{1}^{r_{1}}… q_{s}^{r_{s}}M$, where $r_{1}, …, r_{s}$ are non-negative integers and $M$ is an integer relatively prime to $q_{1}… q_{s}$ . We define the $S$ -part $[m]_{S}$ of $m$ by $[m]_{S}:=q_{1}^{r_{1}}… q_{s}^{r_{s}}$ . Let $(u_{n})_{n\geqslant 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\unicode[STIX]{x1D700}>0$, there exists an integer $n_{0}$ such that $[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$ 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 $(u_{n})_{n\geqslant 0}$, 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 $(u_{n})_{n\geqslant 0}$ and $S$ .