‐PARTS OF TERMS OF INTEGER LINEAR RECURRENCE SEQUENCES

Abstract

Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1^ r1 . . . qs^rs M, where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]_S of m by [m]_S := q1^r1 . . . qs^rs.Let (u_n )_{n≥0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every e > 0, there exists an integer n_0 such that [u_n ]_S ≤ |u_n |^e 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≥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≥0} and S.