Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on the number of squares in recurrence sequences: arbitrary $b$, III

Paul M Voutier

TL;DR

This work extends prior analyses of the number of distinct squares in binary recurrence sequences to general negative norms $N_{\alpha}$, proving that only a small, finite set of squares can occur for large indices. The authors combine hypergeometric approximations, gap principles, and careful casework on the parameter $r_0$ and 4th roots of unity to derive upper bounds and eventual contradictions for configurations with many squares. They show that for $N_{\alpha}<0$ there are at most four distinct squares among suitably large $y_k$, and in several regimes (notably small $b$ or large $d$) the bound tightens to at most nine squares, with explicit lower bounds on $d$ in terms of $b$ and $u$ guiding the analysis. The paper also includes computational verification for removing $d$-bounds in specific small-$b$ cases, and provides public code to support the hypergeometric approach, highlighting the practical impact for Diophantine approximation and Pell-type sequence analysis.

Abstract

We generalise our earlier work on the number of squares in binary recurrence sequences, $\left\{ y_{k} \right\}_{k \geq -\infty}$. In the notation of our previous papers, here we consider the case when $N_α$ is any negative integer and $y_{0}=b^{2}$ for any positive integer, $b$. We show that there are at most $4$ distinct squares with $y_{k}$ sufficiently large. This allows us to also show that there are at most $9$ distinct squares in such sequences when $b=1,2$ or $3$, or once $d$ is sufficiently large.

Bounds on the number of squares in recurrence sequences: arbitrary $b$, III

TL;DR

This work extends prior analyses of the number of distinct squares in binary recurrence sequences to general negative norms , proving that only a small, finite set of squares can occur for large indices. The authors combine hypergeometric approximations, gap principles, and careful casework on the parameter and 4th roots of unity to derive upper bounds and eventual contradictions for configurations with many squares. They show that for there are at most four distinct squares among suitably large , and in several regimes (notably small or large ) the bound tightens to at most nine squares, with explicit lower bounds on in terms of and guiding the analysis. The paper also includes computational verification for removing -bounds in specific small- cases, and provides public code to support the hypergeometric approach, highlighting the practical impact for Diophantine approximation and Pell-type sequence analysis.

Abstract

We generalise our earlier work on the number of squares in binary recurrence sequences, . In the notation of our previous papers, here we consider the case when is any negative integer and for any positive integer, . We show that there are at most distinct squares with sufficiently large. This allows us to also show that there are at most distinct squares in such sequences when or , or once is sufficiently large.

Paper Structure

This paper contains 23 sections, 13 theorems, 93 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Conjectures
  5. Previous Results
  6. New Results
  7. Future work
  8. Lemmas about $\left( x_{k} \right)_{k=-\infty}^{\infty}$ and $\left( y_{k} \right)_{k=-\infty}^{\infty}$
  9. Proof of Theorem \ref{['thm:1.2-seq-new']}
  10. Choice of indices
  11. Further Prerequisites
  12. $r_{0}=1$ and $\zeta_{4}p/q \neq p_{1}/q_{1}$ for all $4$-th roots of unity, $\zeta_{4}$
  13. $r_{0}=1$ and $\zeta_{4}p/q = p_{1}/q_{1}$ for some $4$-th root of unity, $\zeta_{4}$
  14. $r_{0}>1$, $\zeta_{4}p/q \neq p_{r_{0}}/q_{r_{0}}$ for all $4$-th roots of unity, $\zeta_{4}$
  15. $r_{0}>1$ and $\zeta_{4}p/q = p_{r_{0}}/q_{r_{0}}$ for some $4$-th root of unity, $\zeta_{4}$
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $b=1$, $a$ and $d$ be positive integers, where $d$ is not a square, $N_{\alpha}<0$ and $-N_{\alpha}$ is a square. (a) If $u=1$, $t^{2}-du^{2}=-4$, $N_{\alpha} \equiv 12 \pmod{16}$, $\gcd \left( a^{2},d \right)=1,4$ and one of $y_{\pm 1}$ is a perfect square, then there are at most three distinct

Theorems & Definitions (20)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 10 more