Purely periodic Rosen continued fractions (Q6595538)

Hecke groups are kind of a general case of the well-known modular group, where \(m=3\). They are free products of two cyclic groups of orders 2 and m generated by an elliptic element and a parabolic element or two elliptic elements. There are many papers on the algebraic structure, the normal subgroups and the structures of their elements. For small values of \(m\) less than 7, the elements of the Hecke groups are given in terms of continued fractions. For \(Z[\sqrt(D)]\), it has been recently proven that each unit in this ring have Rosen continued fraction expansion. In this paper, the authors study the elements of \(Z[\lambda_m],\ m \in {4, 6}\), whose Rosen continued fractions are purely periodic and give a characterisation.