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A Computational Exploration of Gaussian and Eisenstein Moats

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A Computational Exploration of Gaussian and Eisenstein Moats A Computational Exploration of Gaussian and Eisenstein Moats By Philip P. West A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS IN SCIENCE (MATHEMATICS) at the CALIFORNIA STATE UNIVERSITY CHANNEL ISLANDS 20 14 copyright 20 14 Philip P. West ALL RIGHTS RESERVED APPROVED FOR THE MATHEMATICS PROGRAM Brian Sittinger, Advisor Date: 21 December 20 14 Ivona Grzegorzyck Date: 12/2/14 Ronald Rieger Date: 12/8/14 APPROVED FOR THE UNIVERSITY Doctor Gary A. Berg Date: 12/10/14 Non- Exclusive Distribution License In order for California State University Channel Islands (C S U C I) to reproduce, translate and distribute your submission worldwide through the C S U C I Institutional Repository, your agreement to the following terms is necessary. The authors retain any copyright currently on the item as well as the ability to submit the item to publishers or other repositories. 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The C S U C I Institutional Repository will clearly identify your names as the authors or owners of the submission, and will not make any alteration, other than as allowed by this license, to your submission. Title of Item: A comutational exploration of Gaussian and Eisenstein Moats 3 to 5 keywords or phrases to describe the item: Gaussian Moat, Eisenstein Moat, Eisenstein Prime Density Authors Name (Print): Philip P. West Authors Signature Date: Dec 5, 20 14 This is a permitted, modified version of the Non- exclusive Distribution License from M I T Libraries and the University of Kansas. DEDICATION I dedicate this paper to the following two influential women in my life. First, I acknowledge my wife Sharon Hines- West who has always been supportive of me, both with her encouragement and with her own knowledge of mathemat- ics. Secondly, I am grateful to my mother Doctor Julia S. West, who despite many personal struggles, received her doctorate at age 58 and taught her entire life right up to her passing at age 81. Her example inspired me to pursue this goal later in life. roman numeral 4 ACKNOWLEDGEMENTS This thesis would not have been possible without the support of the Mathe- matics Department at California State University Channel Islands. In partic- ular, I would like to thank my advisors Doctor Brian Sittinger, Ronald Rieger, and Doctor. Ivona Grzegorczyk. I also would like to acknowledge Doctor Jorge Garcia for his help formatting this thesis using Latex. Camarillo, California December 3, 20 14 v ABSTRACT If one imagines the Gaussian primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture. If the frog's journey terminates for a given hop size, it implies that a prime free "moat" greater than the hop size completely surrounds the origin. In the Chauvenet Prize- winning paper "A Stroll Through the Gaussian Primes", Ellen Gethner, Stan Wagon, and Brian Wick [4] explored this problem and by computational methods proved the existence of a square root of 26 -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist. In their concluding remarks, Gethner et al. note that "Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1." 1 This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. By computational methods, we prove the existence of a bounding square root of l6 -moat in the Eisenstein primes and provide an estimate for the bounds of any square root of k -moat. A subtle difference be- tween the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation. 2 TABLE OF CONTENTS 1 Introduction 5 2 Mathematical Background 9 2 point 1 Arithmetic in the Gaussian and Eisenstein Integers 9 2 point 2 The Chinese Remainder Theorem 12 2 point 3 Gaussian Moats (Gethner) 13 2 point 4 Gaussian Prime Density (Vardi) 17 2 point 5 Estimating Bounds on Gaussian Moats (Vardi) 18 3 Computational Background 21 3 point 1 Improved Computational Results (Tsuchimura) 21 3 point 2 Depth-first Search Algorithm 25 3 point 3 Multiple Precision Integers and Rationals Library 26 3 point 4 Miller- Rabin Primality Testing 27 4 Mathematical Results 31 4 point 1 Annular Moats 31 4 point 2 Eisenstein Prime Density 33 3 TABLE OF CONTENTS 4 point 3 Finding p parenthesis k parenthesis for a k- connected Eisenstein Prime Component 35 4 point 4 Estimating Methods for Bounding Eisenstein Moats 37 4 point 5 Generalizing Gethner's Proof to the Eisenstein Integers 39 5 Computational Results 45 5 point 1 Overview of Approach 45 5 point 2 k- Connected Gaussian Prime Algorithm 46 5 point 3 k- Connected Eisenstein Prime Algorithm 50 5 point 4 Gaussian Prime Viewer for Validation Purposes 54 5 point 5 Validation of Gaussian Moat Data 56 5 point 6 Eisenstein Moat Data 57 6 Conclusion 60 7 Appendix 65 7 point 1 Gethner's Proof Generalized for Imaginary Quadratic Integers 65 7 point 2 MAPLE Code for square root of 7-connected E- Prime Plot 71 7 point 3 C- code for Gaussian Prime Viewer 72 7 point 4 C- code for Eisenstein Prime Connected Components 77 Bibliography 87 4 Chapter 1 INTRODUCTION Simple problems often lead to more interesting and difficult questions. For instance, is it possible to travel on the real number line to infinity by taking bounded steps on only the prime numbers? The answer is no and easily seen by showing arbitrarily large prime-free gaps exist on the number line by ob- serving that for any integer n greater than 1, parenthesis n plus 1 parenthesis exclamation mark plus 2, parenthesis n plus 1 parenthesis exclamation mark plus 3, dot dot dot, parenthesis n plus 1 parenthesis exclamation mark plus parenthesis n plus 1 parenthesis are n consecutive composite integers. Now that we know arbitrarily large gaps without primes exist on the real number line, then inquiring minds would ask, "Are there also Gaussian prime-free moats of arbitrarily large width that surround the origin in the complex plane?" The answer is, "Probably, but we have not been able to prove it." 5 A problem more analogous to the question about arbitrary prime free- gaps on the real line is whether two- dimensional prime-free neighborhoods of arbitrary radius k exist. This was proven by Gethner et al. in 19 98, [4]. We provide a direct modification of this proof from the Gaussian integers to the Eisenstein integers. Definition 1 point 0 point 1 (k- connected component) A set of vertices in a graph whose vertices to each other by edges of length k or less. We will only consider the set of vertices representing Gaussian primes or the set of vertices representing Eisen- stein primes in the complex plane. In both cases, our connected component will always start with the element closest to the origin of the complex plane. Gethner et al. provided computational proofs for the existence of Gaussian moats with the specific maximum distance reachable for all connected com- ponents surrounded by moats up to square root of 20. Additionally, they computationally demonstrated that a square root of 26 -moat exists, but only provided a bound for the max- imum reachable distance of its connected component. Sometimes, applied mathematics can provide insights for solving abstract problems. For instance, Percolation Theory, which describes the behavior of connected clusters in a random graph, is used in material science to model 6 movement and filtering of fluids through porous materials.
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