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Binary matrices of order 3 associated with the Pell sequence

Wilson Arley Martinez, Samin Ingrid Ceron

TL;DR

The paper develops a linear-algebraic framework for Pell numbers by employing binary $3\times3$ matrices with determinant $0$, $1$, and $-1$ to generate recurrences, derive Binet-type formulas, and establish Cassini-like identities. It provides explicit matrix representations for Pell and generalized sequences, analyzes diagonalizations to obtain closed-form expressions, and investigates divisibility and Sidon properties of associated recurrences. A key contribution is the complete classification of binary $3\times3$ matrices that realize Pell sequences into three conjugacy classes, demonstrated through computational methods. Together, these results enhance the algebraic understanding of Pell-type sequences and their matrix realizations, with potential applications in number theory and Diophantine contexts.

Abstract

In this paper, we construct Pell matrices, analogous to Fibonacci matrices, to study algebraic properties of Pell numbers via linear algebra. This framework yields identities involving the trace, inverse, and determinant, as well as matrix products that generate recurrence relations and closed-form expressions. Additionally, we classify all binary 3x3 matrices that generate the Pell equation through conjugation, providing a complete characterization of such matrices.

Binary matrices of order 3 associated with the Pell sequence

TL;DR

The paper develops a linear-algebraic framework for Pell numbers by employing binary matrices with determinant , , and to generate recurrences, derive Binet-type formulas, and establish Cassini-like identities. It provides explicit matrix representations for Pell and generalized sequences, analyzes diagonalizations to obtain closed-form expressions, and investigates divisibility and Sidon properties of associated recurrences. A key contribution is the complete classification of binary matrices that realize Pell sequences into three conjugacy classes, demonstrated through computational methods. Together, these results enhance the algebraic understanding of Pell-type sequences and their matrix realizations, with potential applications in number theory and Diophantine contexts.

Abstract

In this paper, we construct Pell matrices, analogous to Fibonacci matrices, to study algebraic properties of Pell numbers via linear algebra. This framework yields identities involving the trace, inverse, and determinant, as well as matrix products that generate recurrence relations and closed-form expressions. Additionally, we classify all binary 3x3 matrices that generate the Pell equation through conjugation, providing a complete characterization of such matrices.
Paper Structure (13 sections, 41 theorems, 304 equations, 1 table)

This paper contains 13 sections, 41 theorems, 304 equations, 1 table.

Table of Contents

  1. Recurrences Generated by Binary $3 \times 3$ Matrices with Determinant Zero
  2. The Matrix Representation of Pell Numbers
  3. Diagonalization of the Generating Matrix and Binet’s Formula
  4. Recurrences Generated by Binary $3 \times 3$ Matrices with Determinant One
  5. The Matrix Representation of Recurrences and Their Identities
  6. Inverse Powers of the Generating Matrix and Identities for Recurrence Relations
  7. Diagonalization of the Generating Matrix and Binet’s Formula
  8. Recurrences Generated by Binary $3 \times 3$ Matrices with Determinant Minus One
  9. The Matrix Representation of Recurrences and Their Identities
  10. Inverse Powers of the Generating Matrix and Identities for Recurrence Relations
  11. Diagonalization of the Generating Matrix and Binet’s Formula
  12. Divisibility Properties and Greatest Common Divisors of Terms in the Recurrence Relation $r_n$
  13. Classification of Binary $3 \times 3$ Matrices Associated with Pell Numbers

Key Result

Lemma 1.1.1

Let $u$ be the matrix Then, where $n\in \mathbb{Z}^{+}$ and $E_{n}$ denotes the Pell numbers, defined by the recurrence relation with initial conditions $E_0 = 0$ and $E_1 = 1$.

Theorems & Definitions (83)

  • Lemma 1.1.1
  • proof
  • Lemma 1.1.2
  • Corollary 1.1.3
  • Corollary 1.1.4
  • proof
  • Theorem 1.2.1
  • proof
  • Theorem 1.2.2
  • proof
  • ...and 73 more