2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129505The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and $α$-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure.15 pagesNumber TheoryDynamical Systems11K50; 37A45text