S-parts of terms of integer linear recurrence sequences

2016/11/02 by Bugeaud, Yann, Evertse, Jan-Hendrik
#11B37 #11J86 #11J87 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1611.00485

Abstract

Let S = \q₁, … , q_s\ be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q₁^r₁ … q_s^r_s M, where r₁, … , r_s are non-negative integers and M is an integer relatively prime to q₁ … q_s. We define the S-part [m]_S of m by [m]_S := q₁^r₁ … q_s^r_s. Let (u_n)_{n ≥ 0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every ε > 0, there exists an integer n₀ such that [u_n]_S≤ |u_n|^ε holds for n > n₀. Our proof is ineffective in the sense that it does not give an explicit value for n₀. 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.

Related

• S-parts of sums of terms of linear recurrence sequences
• Intervals without primes near an iterated linear recurrence sequence
• Sums of S-units in sum of terms of recurrence sequences
• Quantitative growth of linear recurrences
• Integers representable as differences of linear recurrence sequences