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$n$-defective Lehmer pairs for small $n$: Corrections and Clarifications

Paul M. Voutier

TL;DR

The paper corrects and clarifies the classification of $n$-defective Lehmer pairs for small indices $n\in\{3,4,5,6,8,10,12\}$, providing explicit $(a,b)$-parametrizations that generate all non-equivalent Lehmer pairs. It uncovers a previously omitted example for $n=5$ and supplies detailed corrections to earlier tables and index ranges in Ab and BHV, including careful handling of equivalence and degeneracies. The authors extend necessary auxiliary sequences to negative indices and implement computational checks with PARI/GP to verify completeness up to substantial bounds. The results refine the understanding of Lehmer sequence behavior at small indices and ensure robust frameworks for applications relying on primitive divisors in binary recurrence sequences.

Abstract

We provide some corrections and clarifications of statements in \cite{Ab} and \cite{BHV} regarding elements in Lehmer sequences that have no primitive divisors for small $n$.

$n$-defective Lehmer pairs for small $n$: Corrections and Clarifications

TL;DR

The paper corrects and clarifies the classification of -defective Lehmer pairs for small indices , providing explicit -parametrizations that generate all non-equivalent Lehmer pairs. It uncovers a previously omitted example for and supplies detailed corrections to earlier tables and index ranges in Ab and BHV, including careful handling of equivalence and degeneracies. The authors extend necessary auxiliary sequences to negative indices and implement computational checks with PARI/GP to verify completeness up to substantial bounds. The results refine the understanding of Lehmer sequence behavior at small indices and ensure robust frameworks for applications relying on primitive divisors in binary recurrence sequences.

Abstract

We provide some corrections and clarifications of statements in \cite{Ab} and \cite{BHV} regarding elements in Lehmer sequences that have no primitive divisors for small .

Paper Structure

This paper contains 22 sections, 1 theorem, 9 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Results
  4. Changes from AbBHV
  5. $n=3$ changes
  6. $n=4$ changes
  7. $n=5$ changes
  8. $n=6$ changes
  9. $n=8$ changes
  10. $n=10$ changes
  11. $n=12$ changes
  12. Code
  13. Proof of Theorem \ref{['thm:main']}
  14. $n=3$
  15. $(a,b)=(1+q,1-3q)$
  16. ...and 7 more sections

Key Result

Theorem 1

For $n \in \left\{ 3,4,5,6,8,10,12 \right\}$, up to equivalence, all $n$-defective Lehmer pairs are of the form $\left( \left( \sqrt{a}-\sqrt{b} \right)/2, \left( \sqrt{a}+\sqrt{b} \right)/2 \right)$, where $(a,b)$ is given in Table table:lehmer with $k, \ell, q \in \mathbb{Z}$ and $\varepsilon = \p

Theorems & Definitions (1)

  • Theorem 1