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VIDEO LIBRARY |
À.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
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On Lagrange algorithm for reduced algebraic irrationalities N. M. Dobrovol'skii Tula State Pedagogical University |
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Abstract: In 2015, we proved that the calculation of another successive coefficient of the continued fraction expansion of a given algebraic number Since all the roots are different, we introduce the following notation $$ \delta(\alpha)=\min_{2\le j\le n}\left|\alpha^{(1)}-\alpha^{(j)}\right|>0, $$ Theorem. Let $$ \alpha\,=\,\alpha_{0}\,=\,q_{0}\,+\,\cfrac1{q_{1}\,+\,\cfrac{1}{\ddots+\cfrac{1}{q_{k}+\cfrac{1}{\ddots}}}}\,. $$ Define the index \begin{equation*} \frac{2(n-1)}{Q_{m_0-1}\delta(\alpha) }\,<\,\varepsilon. \end{equation*} Then the relations: $$ (-1)^{m}f_{0}\left(\frac{q_{m}^{*}P_{m-1}+P_{m-2}}{q_{m}^{*}Q_{m-1}+Q_{m-2}}\right)\!>\!0\quad \text{and}\quad (-1)^{m}f_{0}\left(\frac{(q_{m}^{*}+1)P_{m-1}+P_{m-2}}{(q_{m}^{*}+1)Q_{m-1}+Q_{m-2}}\right)\!<\!0, $$ $$ (-1)^mf_{0}\left(\frac{(q_{m}^{*}+1)P_{m-1}+P_{m-2}}{(q_{m}^{*}+1)Q_{m-1}+Q_{m-2}}\right)>0, $$ $$ (-1)^mf_{0}\left(\frac{q_{m}^{*}P_{m-1}+P_{m-2}}{q_m^*Q_{m-1}+Q_{m-2}}\right)<0, $$ hold for any \begin{equation*} q_{m}^{*}\,=\,\left[\frac{f'_{0}\left(\frac{P_{m-1}}{Q_{m-1}}\right)}{Q_{m-1}^2 \left|f_{0}\left(\frac{P_{m-1}}{Q_{m-1}}\right)\right|}-\frac{Q_{m-2}}{Q_{m-1}}\right]. \end{equation*} Language: English |