Constructing families of long continued fractions. (Q5943433)

Many authors exhibited a lot families of parametric quadratic integers \(\sqrt{M}\) for which the continued fraction expansion can be explicitly exhibited with lengths depending on some parameter \(k\), hence with lengths going to infinity. In the paper under review, D. J. Madden obtains new families of such quadratic integers \(\sqrt{M}\). His method for producing them is new and interesting and is based on matrix identities involving the classical matrices \(\left(\begin{smallmatrix} 0 & 1\\ 1 & t\end{smallmatrix}\right)\) which occur in the theory of continued fractions.