Aspects of Fibonacci numbers

Abstract

This thesis consists of two parts. The first part, which is Chapter 2, is a survey on some aspects of Fibonacci numbers. In this part, we tried to gather some interesting properties of these numbers and some topics related to the Fibonacci sequence from various references, so that the reader may get an overview of the subject. After giving the basic concepts about the Fibonacci numbers, their arithmetical properties are studied. These include divisibility and periodicity properties, the Zeckendorf Theorem, Fibonacci trees and their relations to the representations of integers, polynomials used for deriving new identities for Fibonacci numbers and Fibonacci groups. Also in Chapter 2, natural phenomena related to the golden section, such as certain plants having Fibonacci numbers for the number of petals, or the relations of generations of bees with the Fibonacci numbers are recounted. In the second part of the thesis. Chapter 3, we focused on a Fibonacci based random number sequence. We analyzed and criticized the generator Sfc = k(j>—[k(j)] by applying some standart tests for randomness on it. Chapter 5, the Appendix consists of Fortran programs used for executing the tests of Chapter 3.