On periodic sequences for algebraic numbers

In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers which are the leglengths of a Pythagorean triangle. Using the idea of Pythagorean rationals, we generate two families of rational triangles. We define a rational triangle to be a triangle with rational sidelengths and area.

We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions leads to the visualization of special subfamilies of continued fractions with torus triangulations (i.e. combinatorics of their fundamental domains) that possess explicit regularities.Several cases of such subfamilies are studied in detail;...

In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in $\mathbb{Q}_p$ that have a periodic expansion with period of length four, solving...

Let $Q(\alpha)$ be the simplest cubic field, it is known that $Q(\alpha)$ can be generated by adjoining a root of the irreducible equation $x^{3}-kx^{2}+(k-3)x+1=0$, where $k$ belongs to $Q$. In this paper we have established a relationship between $\alpha$, $\alpha'$ and $k,k'$ where $\alpha$ is a root of the equation $x^{3}-kx^{2}+(k-3)x+1=0$ and $\alpha'$ is a root of the same equation with $k$ replaced by $k'$ and $Q(\alpha)=Q(\alpha')$.

We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction expansion for $(1+t)^r$, $r\in\mathbb{Q}$ in 1813. As an application, we apply an analogue of the hypergeometric method to one of those families and derive non-trivial lower bounds on the distance $|x - \frac{p}{q}|$ between one of the real roots ...

To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for integers to spaces with more than three dimensions. Thus, taking up a definition of polygonal numbers given by Diophantus and by Nicomachus, and generalizing the Pythagorean concept of gnomon, one is led through quite elementary means to a s...

We define an algorithm which begins with an sequence of sequences, and produces a single sequence, with following property: If at least one of the original sequences has a tail that is periodic, then the output sequence has a periodic tail, and conversely. Our purpose is to supplement a result by Dasaratha et al., in which a real number input results in a countable family of sequences, with the property that at least one is eventually periodic if and only if the input is a cu...

We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give, for each of our algorithms, the complete characterization of elements having purely periodic expansions.

We prove that there exist infinitely many algebraic integers with continued fraction expansion of the kind $[a_0, \overline{a_1, \ldots, a_n, s}]$ where $(a_1, \ldots, a_n)$ is a palindrome and $s \in \mathbb N_{\geq1}$, characterizing all the algebraic integers with such expansions. We also provide an explicit method for finding $s$ and determining the corresponding algebraic integer. Moreover, we deal with the particular case $(a_1, \ldots, a_n) = (m, \ldots, m)$ describing...

We study a new connection between multidimensional continued fractions, such as Jacobi--Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposab...