S-parts of sums of terms of linear recurrence sequences

Abstract

Let \(S= \{ p_1, \ldots, p_s\}\) be a finite, non-empty set of distinct prime numbers and \((U_{n})_{n \geq 0}\) be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and \(w = (w_k, \ldots, w_1)\in\mathbb{Z}^k, w_1, \ldots, w_k\neq 0\) we define \((U_j^{(k, w)})_{j\geq 1}\) an increasing sequence composed of integers of the form \(|w_kU_{n_k} +\cdots + w_1U_{n_1}|\), \( n_k>\cdots >n_1\). Under certain assumptions, we prove that for any \(\varepsilon >0\), there exists an integer \(n_{0}\) such that \([U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},\) \({\rm for}\, j > n_0\), where \([m]_S\) denotes the S-part of the positive integer m. On further assumptions on \((U_{n})_{n \geq 0}\), we also compute an effective bound for \([U_j^{(k, w)}]_S\) of the form \((U_j^{(k,w)})^{1-c}\), where c is a positive constant depending only on r, \(a_1\) , . . ., \(a_r\), \(U_0\), . . ., \(U_{r-1}\) , \(w_1\), . . ., \(w_k\) and S.