It is not known whether or not the Jacobi-Perron Algorithm of a vector in Rn_x, n > 3, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields. The main result of this paper is expressed

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... in the following theorem: 3 There are infinitely many real cubic fields Q(w), w cubefree, a and T natural 2 numbers, such that the Jacobi-Perron Algorithm of the vector (w, w ) becomes periodic; the length of the primitive preperiod is four, the length of the primi-