Related papers: Fibonacci Sequences And Real Quadratic p-Rational …
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…
Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and…
In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~\(f\) to have exactly…
We investigate general properties of number sequences which allow explicit representation in terms of products. We find that such sequences form whole families of number sequences sharing similar recursive identities. Restricting to the…
In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.
In this note we study some sequences whose ratio converges to the square root of rationals. Further we analyze some related sequences obtained when the above mentioned ratio simplifies.
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…
Consider an algebraic function like $F(x) = \sqrt{x^3 - 1}$. If $p \in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of…
There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…
We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. In the present paper, we construct the Hilbert genus…
Some aspects of analysis involving fields with absolute value functions are discussed, which includes the real or complex numbers with their standard absolute values, as well as ultrametric situations like the p-adic numbers.
In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which…
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from…
The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with…
This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.
The regularized product of the Fibonacci numbers is evaluated.