Related papers: Fibonacci Numbers, Golden Section and Applications
Inspired by the ancient spiral constructed by the greek philosopher Theodorus which is based on concatenated right triangles, we have created a spiral. In this spiral, called \emph{Fibonacci--Theodorus}, the sides of the triangles have…
This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote…
One source of beauty in mathematics is totally unexpected connections between two fundamentally different objects. For instance, is it not surprising that the time period of a real simple pendulum is linked with a function arising out of…
In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were…
Expressions of type $(p^q-1)/(p-1)$ and $a^2+ab+b^2$, where $a, b$ are natural and $p, q$ are prime numbers, are studied.
Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of…
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.
Two new generalized Fibonacci number summation identities are stated and proved, and two other new generalized Fibonacci number summation identities are derived from these, of which two special cases are already known in literature.
We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.
The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…
Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly…
In this paper we provide some applications of the norm form in some quaternion division algebras over rational field and we give some properties of Fibonacci sequence and Fibonacci sequence in connection with quaternion elements. We define…
Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.
Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of…
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known…
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in…
One possible data encryption scheme is related to stream ciphers, which use a sufficiently long pseudo-random sequence. To increase the cryptographic strength of the cipher, linear shift algorithms (generated by linear recurrent sequences…