Nonuniqueness Properties of Zeckendorf Related Decompositions Open Access

Abstract

Zeckendorf’s Theorem states that every natural number can be uniquely written as the sum of distinct and nonconsecutive terms of the Fibonacci number sequence. Similarly, every natural number can be written as the sum of distinct and nonconsecutive terms of the Lucas number sequence. Although such decompositions of natural numbers in the Lucas number sequence need not be unique, there has been much progress on categorizing those natural numbers that do not carry this uniqueness property. We investigate the proportion of natural numbers that cannot be uniquely written as the sum of distinct and nonconsecutive terms of the Lucas number sequence. In doing so, we show the limiting value of this proportion and speculate on future research that generalizes the ideas presented in this paper.

Table of Contents

1 Introduction - 1

2 Motivation - 5

3 Preliminaries and Background - 9

4 Main Results - 13

5 Future Work - 21

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Number Theory
Committee Chair / Thesis Advisor	David Zureick-Brown, Emory University
Committee Members	Shanshuang Yang, Emory University Elena Glazov-Corrigan, Emory University

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