On a periodic Jacobi–Perron type algorithm

Abstract In 1848 Ch. Hermite formulated a question on periodic representation of cubic numbers. In this paper we introduce a new Jacobi–Perron type algorithm that provides periodicity for all cubic irrationalities in the input and prove the periodicity. This is the first subtractive algorithm for which the algebraic periodicity is proven. The algorithm is designed for the totally real case, providing the answer to Hermite’s original question in this case.