**Authors:** N. M. Dobrovol’skii, I. N. Balaba, I. Yu. Rebrova, N. N. Dobrovol’skii

### 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.\\
\hspace*{6mm}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.\\
\hspace*{6mm}The definition of generalized Pisot numbers differs from the definition of Pisot numbers
by absence of the requirement to be integer.

Tula State Lev Tolstoy Pedagogical University

Lenina prospect, 125, 300026, Tula, Russia

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