On Lagrange algorithm for reduced algebraic irrationalities

N. M. Dobrovol'skii ; I. N. Balaba ; I. Yu. Rebrova ; N. N. Dobrovol'skii

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2016), pp. 27-39.

Résumé

In this paper the properties of Lagrange algorithm for expansion of algebraic number are refined. It has been shown that for reduced algebraic irrationalities the quantity of elementary arithmetic operations which needed for the computation of next incomplete quotient does not depend on the value of this incomplete quotient. It is established that beginning with some index all residual fractions for an arbitrary reduced algebraic irrationality are the generalized Pisot numbers. An asymptotic formula for conjugate numbers to residual fractions is obtained. The definition of generalized Pisot numbers differs from the definition of Pisot numbers by absence of the requirement to be integer.

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