New Generalized Continued Fractions for the Multiple Vectors of Extremal Forms

V. I. Parusnikov


Abstract: Davenport and Swinnerton-Dyer found the first 20 extremal thernar cubic forms g_i, the meaning of which is the same as the meaning of the Markov forms for binary quadratic case. Bruno proposed the new generalization of continued fraction expansion algorithm. This algorithm gives the best result with respect to the known algorithms for the multiple vectors connected with some of the first forms g_i. Here we study the periodicity property of the new Bruno algorithm for multiple vectors of all of the Swinnerton-Dyer's forms g_i . We constructed 60 cases: in 25 cases we found a period, in 28 cases a period was not found and in 7 cases the algorithm does not work.