Positive factors of Wythoff matrices
type: Item
instance of: scholarly article
title: Positive factors of Wythoff matrices
zbMATH Open document ID: 0859.15003
DOI: 10.1007/BF02574143
author: Kenneth B. Stolarsky
published in: Semigroup Forum
publication date: 1996-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/135491
review text: Let \(a(n)= \lfloor n \varphi \rfloor\) and \(b(n) = \lfloor n \varphi^2 \rfloor\) where \(\varphi = {\sqrt 5+1 \over 2}\). The \(n\)th Wythoff matrix is \(\sigma(n) = \left(\begin{smallmatrix} n & a(n) \\ a(n) & b(n) \end{smallmatrix} \right)\). Let \({\mathcal S} = \{\sigma(n):n \geq 1\}\) be the set of Wythoff matrices. \({\mathcal S}\) is a multiplicative subsemigroup of \({\mathcal M}^+ ={\mathcal M}^+ (2)\), the semigroup of all nonsingular \(2 \times 2\) matrices over the positive integers \(\mathbb{Z}^+\). It is shown that the factors of \({\mathcal S}\) contain \({\mathcal M}^+\). A similar result is also established relative to \({\mathcal M}^+_k\), the set of all matrices of \({\mathcal M}^+\) congruent to the \(2 \times 2\) identity matrix \(I\) modulo the positive integer \(k\). Some open problems are also discussed.
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 943668
zbMATH Keywords: positive factors
zbMATH Keywords: golden mean
zbMATH Keywords: multiplicative semigroup
zbMATH Keywords: Wythoff matrix
reviewed by: Giacomo Bonanno
MaRDI profile type: Publication
cites work: Q3944542
cites work: Iterated Floor Function, Algebraic Numbers, Discrete Chaos, Beatty Subsequences, Semigroups
cites work: Characterization of the set of values \(f(n)=[n \alpha ], n=1,2,\dots \)
cites work: Q3351453
cites work: The almost PV behavior of some far from PV algebraic integers
cites work: Fibonacci multiplication
cites work: Asymmetric Tent Map Expansions. I. Eventually Periodic Points
cites work: Asymmetric tent map expansions. II: Purely periodic points
cites work: Q3481787
cites work: Half-Silvered Mirrors and Wythoff's Game
cites work: Wythoff pairs as semigroup invariants
cites work: Beatty Sequences, Continued Fractions, and Certain Shift Operators
description: scientific article; zbMATH DE number 943668