Positive factors of Wythoff matrices
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.
scientific article; zbMATH DE number 943668
scientific article