Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twisted rational zeros of linear recurrence sequences

Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, James Worrell

Abstract

We introduce the notion of a twisted rational zero of a non-degenerate linear recurrence sequence (LRS). We show that any non-degenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that $1/3$ and $-5/3$ are the only twisted rational zeros which are not integral zeros.

Twisted rational zeros of linear recurrence sequences

Abstract

We introduce the notion of a twisted rational zero of a non-degenerate linear recurrence sequence (LRS). We show that any non-degenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that and are the only twisted rational zeros which are not integral zeros.
Paper Structure (24 sections, 29 theorems, 109 equations)

This paper contains 24 sections, 29 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Twisted zeros and $p$-adic orders
  3. Finiteness
  4. Preliminaries
  5. Fields
  6. Linear recurrence sequences
  7. Twisted rational zeros and the $p$-adic order
  8. $p$-adic analytic functions
  9. $p$-adic analytic interpolation of a linear recurrence sequence
  10. Proof of Theorems \ref{['theasy']}, \ref{['thtwint']} and \ref{['thtwrat']}
  11. Finiteness of twisted rational zeros
  12. Powers in fields
  13. Equations in roots of unity
  14. A Kummer property
  15. Proof of Theorem \ref{['thfin']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $a$ be a zero of a ${\mathbb Q}$-valued LRS $U$, and $p$ a regular prime for $U$. Then there exist a positive integer $Q$, a positive integer $\kappa$ and an integer $\tau$ such that

Theorems & Definitions (54)

  • Theorem 1.1
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.8: Skolem-Mahler-Lech
  • Theorem 1.9
  • Theorem 1.10
  • Lemma 2.1
  • Remark 2.2
  • ...and 44 more