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 Bul. Acad. Ştiinţe Repub. Mold. Mat.: Year: Volume: Issue: Page: Find

 Bul. Acad. Ştiinţe Repub. Mold. Mat., 2016, Number 2, Pages 27–39 (Mi basm423)

On Lagrange algorithm for reduced algebraic irrationalities

N. M. Dobrovol'skiia, I. N. Balabaa, I. Yu. Rebrovaa, N. N. Dobrovol'skii

a Tula State Lev Tolstoy Pedagogical University, Lenina prospect, 125, 300026, Tula, Russia

Abstract: 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.

Keywords and phrases: minimal polynomial, reduced algebraic irrationality, generalized Pisot number, residual fractions, continued fractions.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-01540a This research was supported by Russian Foundation for Basic Research, grant № 15-01-01540a.

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MSC: 11J17
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Citation: N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, N. N. Dobrovol'skii, “On Lagrange algorithm for reduced algebraic irrationalities”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2016, no. 2, 27–39

Citation in format AMSBIB
\Bibitem{DobBalReb16} \by N.~M.~Dobrovol'skii, I.~N.~Balaba, I.~Yu.~Rebrova, N.~N.~Dobrovol'skii \paper On Lagrange algorithm for reduced algebraic irrationalities \jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat. \yr 2016 \issue 2 \pages 27--39 \mathnet{http://mi.mathnet.ru/basm423}