Video library: O. N. German, Cyclic palindromes and periodic continued fractions

VIDEO LIBRARY


Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications January 30, 2016 12:30, Dorodnitsyn Computing Centre, Department of Mechanics and Mathematics of Lomonosov Moscow State University., 119991, GSP-1, Moscow, Leninskie Gory, 1, Main Building, Department of Mechanics and Mathematics, 16 floor, Lecture hall 16-10

Cyclic palindromes and periodic continued fractions O. N. German Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract: The talk is based on the results of a joint paper with I.A. Tlyustangelov. Since the times of Lagrange it has been known that for each rational $r>1$ different from a perfect square we have the following expansion into continued fraction: $$ \sqrt{r}\,=\,\bigl[a_0;\overline{a_{1},a_{2},\ldots,a_{2},a_{1},2a_{0}\mathstrut}\,\bigr]. $$ Particularly, a period of this continued fraction read back to front is again a period. We call such periods cyclic palindromic and prove the following criterion. Theorem. The continued fraction of a quadratic irrationality $\alpha$ has a cyclic palindromic period if, and only if one of the following statements holds: $(1)$ $\alpha\sim\beta:\ \beta^2\in\mathbb{Q}$; $(2)$ $\alpha\sim\beta:\ (\beta-1/2)^2\in\mathbb{Q}$; $(3)$ $\alpha\sim\beta:\ \beta\bar\beta=1$; $(4)$ $\alpha\sim\beta:\ \beta\bar\beta=-1$. Moreover, $(2)$ is equivalent to $(3)$.