Fibonacci Identities Research Articles

On the Construction of 3D Fibonacci Spirals

The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci spiral is built in 2D starting from a quadratic Fibonacci identity. This made us realize that for any cubic identity there are many different ways of packing cuboids, while only an insignificant fraction of those possible tilings might allow a smooth spiral-like curve to be drawn through them. After reviewing the state of the art, we present accurate details on ways to construct such 3D spirals using affine maps. We go on to prove the continuity and smoothness of such 3D spirals by giving a parametrization of the intersection of the surfaces that define the curves. Throughout the paper, we visualize the resulting 3D spirals by generating geometrically correct stereoscopic views. Finally, it is to be mentioned that the recursive 3D packing of cuboids tends to lead to fractal structures, which will need further investigations.

Arctangent Series on Pell and Pell-Lucas Polynomials via Cassini-Like Formulae

By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.

Identities for Fibonacci and Lucas numbers

In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.

A weighted extension of Fibonacci numbers

We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the q-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the q-case, by Garrett. Some identities are proved by telescoping.

The geometry of some Fibonacci identities in the Hosoya triangle

The geometry of some Fibonacci identities in the Hosoya triangle

Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions

Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.

Some Identities on Sums of Finite Products of the Pell, Fibonacci, and Chebyshev Polynomials

Objectives: This study will introduce some new identities for sums of finite products of the Pell, Fibonacci, and Chebyshev polynomials in terms of derivatives of Pell polynomials. Similar identities for Fibonacci and Lucas numbers will be deduced. Methods: Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations. Findings: In terms of derivatives of Pell polynomials, identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of third and fourth kinds are obtained. Novelty: Existing research has identified identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kinds in terms of derivatives of Fibonacci polynomials or Chebyshev polynomials; identities on sums of finite products in terms of Pell polynomials, however, have not been investigated, so identities primarily in terms of Pell polynomials are obtained. Keywords: Fibonacci Numbers; Pell Numbers; Lucas Numbers; Pell Polynomials; Chebyshev Polynomials

The conumerator and the codenominator

In this paper, we answer the question: “what is the qth Fibonacci number, where q is a positive rational?”. The answer is the codenominator function, which is an integral-valued map. It is defined via a system of functional equations, analogous to a system satisfied by the numerator function. Many Fibonacci identities in the literature can be generalized to the codenominator. It is related to Bird's tree and the involution Jimm of the reals induced by the outer automorphism of PGL(2,Z). The “corational” function Jimm satisfies a set of modular functional equations and maps quadratic irrationals to quadratic irrationals in a non-trivial manner. Although having jumps at rationals, it has a derivative which vanishes almost everywhere. We give a formula which gives the amount of jump of Jimm in terms of the codiscriminant function, which is defined in terms of the codenominator.

Survey of Cubic Fibonacci Identities When Cuboids Carry Weight

The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter of mathematical curiosity, but also motivated by a quest for 3D Fibonacci spirals.
 As we were not able to find any survey on the particular topic of cubic Fibonacci identities we decided to try to fill this void. We started by surveying many Fibonacci identities and recording cubic ones. Obviously, tracing all Fibonacci identities (for identifying a handful) is a daunting task. Checking several hundred we have realized that it is not always clear who the author is. The reason is that in many cases an identity was stated in one article (sometimes without a proof, e.g., as an open problem, or a conjecture) while later being proven in another article, or effectively rediscovered independently by other authors. However, we have done our best to present the identities chronologically. We have supplied our own proof for one identity, having tried, but failed, to find a published proof. For all the other identities, we either proved them on a computer, or else verified by hand their original published proofs. Somehow unexpectedly, our investigations have revealed only a rather small number of cubic Fibonacci identities, representing a tiny fraction of all published Fibonacci identities (most of which are linear or quadratic). Finally, out of these, only a handful of cubic Fibonacci identities are homogeneous.

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

Formulated New Identities of Fibonacci Numbers with James Abacus Diagram

ABSTRACT This paper focuses on developing a strategy to represent a new Fibonacci identities using James abacus. The idea of Graphical chain with respect to the beta numbers and abacus was established to count the number of ways to tile a James diagram chain inscribed in finite grid. Then, the idea of chain abacus and beta numbers were used to find new identities of Fibonacci number

A SHORT NOTE ON FIBONACCI NUMBERS IN A GENERALIZED PYTHAGOREAN TRIPLES

This paper aims to construct a new formula that generates a Fibonacci numbers in a generalized Pythagorean triples. In addition, the paper formulates some Fibonacci identities and discuss some important findings.

Fibonacci Identities from Jordan Identities

Fibonacci Identities from Jordan Identities

More Identities for Fibonacci and Lucas Octonions

In this paper, we give a new approach to obtain identities for Fibonacci Lucas octonions. ..........................................................................................................................................................................................................................................................................................................................................................................................................................................

Gelin-Cesáro identities for Fibonacci and Lucas quaternions

Abstract To date, many identities of different quaternions, including the Fibonacci and Lucas quaternions, have been investigated. In this study, we present Gelin-Cesáro identities for Fibonacci and Lucas quaternions. The identities are a worthy addition to the literature. Moreover, we give Catalan’s identity for the Lucas quaternions.

Some High Degree Generalized Fibonacci Identities

Some High Degree Generalized Fibonacci Identities

More Identities for Fibonacci and Lucas quaternions

In this paper, we define the associate matrix as% \begin{equation*} F=\left( \begin{array}{cc} 1+i+2j+3k & i+j+2k \\ i+j+2k & 1+j+k% \end{array}% \right) . \end{equation*}% By the means of the matrix $F,$ we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities that given in the papers \cite% {halici} and \cite{akyigit2}.

Unified Tiling Proofs of a Family of Fibonacci Identities

Unified Tiling Proofs of a Family of Fibonacci Identities

On a four-parameter generalization of some special sequences

We introduce a new four-parameter sequence that simultaneously generalizes some well-known integer sequences, including Fibonacci, Padovan, Jacobsthal, Pell, and Lucas numbers. Combinatorial interpretations are discussed and many identities for this general sequence are derived. As a consequence, a number of identities for Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Narayana numbers as well as some of their generalizations are obtained. We also present the Cassini formula for the new sequence.

Inverse hyperbolic summations and product identities for Fibonacci and Lucas numbers

In this article we derive summation and product identities for Fibonacci and Lucas numbers. The results have a general character and contain some of the existing results as special cases. Mathematics Subject Classification: 11B39, 11Y60