Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved problems connected to them. In this paper we review the history of the conjecture that the only perfect powers in Fibonacci sequence are 1, 8, and 144. Afterwards we consider more stronger conjecture an...

In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself. In the second part we consi...

While Fibonacci numbers can quite easily be classified as a complete sequence, they have the unusual property that a particular explicitly defined subset of such sequences is complete and also gives unique representations of natural numbers. This can be extended to all integers via the ”nega-Fibonacci” numbers. Further, an unusual binary operation on the natural numbers may be defined and prove...

Let F n be the nth Fibonacci number. The order of appearance zn of a natural number n is defined as the smallest natural number k such that n divides F k. For instance, for all n F m ≥ 5, we have zn − 1 > zn < zn 1. In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.

Herein we experimentally study magnetic multilayer metamaterials with broken translational symmetry. Epitaxially-grown iron-gold (Fe-Au) multilayers modulated using Fibonacci sequence-referred to as magnetic inverse Fibonacci-modulated multilayers (IFMs)-are prepared using ultra-high-vacuum vapor deposition. Experimental results of in-situ reflection high-energy electron diffraction, magnetizat...

The purpose of this paper is to study properties of mapped shuffled Fibonacci languages F(afi) and F^vy Let X — {a?6} be an alphabet and let X* be the free monoid generated by X. Let 1 be the empty word and let X+ — X* \ {1}. The length of a word u is denoted by lg(w). Every subset of X* is called a language. For two words u, v E X + ? we consider the following type of Fibonacci sequence of words:

This paper is concernedwith the analysis of theworst case behavior of Hopcroft’s algorithm for minimizing deterministic finite state automata. We extend a result of Castiglione, Restivo and Sciortino. They show that Hopcroft’s algorithm has a worst case behavior for the automata recognizing Fibonacci words. In a previous paper, we have proved that this holds for all standard Sturmian words havi...

The purpose of the present paper is to prove that there are finitely many binomial coefficients of the form (f in certain binary recurrences, and give a simple method for the determination of these coefficients. We illustrate the method by the Fibonacci, the Lucas, and the Pell sequences. First, we transform both of the title equations into two elliptic equations and apply a theorem of Mordell ...

In a recent paper [1] , J. L. Brown, Jr., and R. L, Duncan showed that the sequence *{ QnFn I is uniformly distributed modulo 1 (u.d. mod 1), where c/7 denotes the natural logarithm and Fn is the/? Fibonacci number. In this paper we show that some modifications of these ideas have some interesting consequences concerning the distribution of the first digits of the Fibonacci numbers. This also a...