Common values of a class of linear recurrence

2021/11/22 by Attila Pethő, Pethő, Attila
Computer Science · Mathematics · #11B37 #11D61 #11D75 #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2111.11081

openalex publication_date 2021/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (a_n), (b_n) be linear recursive sequences of integers with characteristic polynomials A(X),B(X)∈ ℤ[X] respectively. Assume that A(X) has a dominating and simple real root α, while B(X) has a pair of conjugate complex dominating and simple roots β,β. Assume further that α/ β and β/β are not roots of unity and δ= log |α|/ log |β| ∈ ℚ. Then there are effectively computable constants c₀,c₁>0 such that the inequality |a_n - b_m| gt; |a_n|^{1-(c₀ log2 n)/n} holds for all n,m ∈ ℤ²_{≥ 0} with max\n,m\>c₁.

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