On a family of generalized continued fraction expansions with periodic length going to infinity (Q1896581)

C. Levesque and G. Rhin exhibited a family of cubic fields \(\mathbb{Q} (\omega )\), \(\omega^3 - c^m \omega^2 - (c - 1) \omega - c^m\) \((c,m \in \mathbb{N}, c \geq 2)\), for which the Jacobi algorithm of \((\omega^2 - c^m \omega - c + 1, \omega)\) is purely periodic of length \(3m + 1\), thereby exhibiting an explicit unit through the Hasse-Bernstein formula. The author uses the Voronoï algorithm to show that this unit is fundamental in the order \(\mathbb{Z} (\omega)\) by writing down explicitly the period (again of length \(3m + 1)\) of this Voronoï algorithm. His results can be compared to those of \textit{Farhane-Dubois} (asymptotic results) [see Zbl 0837.11006 below] and those of \textit{B. Adam} (Thesis, Metz).