We prove that, for any unit in a real number field of degree , there exits only a finite number of n-tuples in which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for . For we give an explicit algorithm to compute all these pairs.
We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex-...