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_________________________________________________________________________________________ Copyright 2019 All rights reserved. No part of this book may be used or reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher. This work was printed in the United States. _________________________________________________________________________________________ Cover Page: Worradirek/Shutterstock.com Beitler, Kenneth W. Using the Greatest Integer Function to reveal the magic in Number Theory and Analysis Includes bibliographical references and index 1. United States-Theoretical and Applied Mathematics, I. Title, February 2019. _________________________________________________________________________________________ I dedicate this work to everyone who shares the uncommon goal of realizing the intricate beauty of mathematics. __________________________________________________________________________________________ Table of Contents _______________ _ Page Preface: An introduction to this text..................................3 PART I Introduction Chapter 1: An Introduction to the Greatest Integer Function..............6 1.1: Rounding, Truncation, and Transformations 1.2: An Analysis of the Greatest Integer Function Chapter 2: More Properties of the Greatest Integer Function.............19 2.1: Elementary Arithmetic Functions 2.2: Congruence Relations 2.3: Summation and Integration PART II Applications Chapter 3: Number Bases and Digits......................................30 3.1: The Definitions and Properties of the Decimal Digit 3.2: Using Digits for Change of Number Base Operations Chapter 4: The Division Algorithm and Reciprocity.......................38 4.1: Reciprocal Subtraction, the Euclidean Approach 4.2: Quadratic Reciprocity 4.3: Dedekind Reciprocity Chapter 5: Miscellaneous Topics in Number Theory and Discrete Math......52 5.1: Finding Integer Solutions to One−Variable Equations 5.2: Graphics and Block−Truncation of Images 5.3: An introduction to Analytic Number Theory Chapter 6: Advanced Series Acceleration Methods.........................65 6.1: Series Acceleration for Monotone Decreasing Sequences 6.2: Fourier series Acceleration Appendix.: Part III Chapter Previews, References and Index..............74 7.1: An Introduction to ODEs with Piecewise Constant Arguments 8.1: General methods for solving Diophantine Equations Preview PREFACE This work can best be described as a study of various integer-intensive topics in mathematics. It is currently an exotic blend of discrete math, number theory and analysis. The topics are chosen in part based on their practical importance or theoretical interest. But our main criterion for selection is that our ability to study them in a rigorous and straightforward manner can be improved using the greatest integer function (or integer functions that can be easily expressed in terms of the greatest integer function). In this respect, our approach is unique for many of these topics and it has produced some original results. Although most of the material (and all the material in Part II) is not specifically about the greatest integer function, we need to study the greatest integer function first so that we can better apply its properties and basic applications to the study of these topics. What is the greatest integer function? The greatest integer of x is usually denoted as [x] and it represents the greatest whole number less than or equal to x. For example, [2.00]=2, [2.99]=2, and [−3.14]=−4. This function and its square bracket notation were first introduced by the German mathematician and physicist Johann Carl Friedrich Gauss in 1808 in his third proof of quadratic reciprocity, which is presented in Chapter 4. This function is also known as the floor function, bracket function, or step function, and it is sometimes denoted as x, [[x]], E(x), or int(x). Aside from using int(x) in computations on the TI−83 graphing calculator, this work does not use any alternative name or notation for the greatest integer function in any context. Conversely, wherever square brackets are used throughout this work, the reader may assume that they denote the greatest integer function unless otherwise specified. The advantage to using the square bracket notation is twofold in that 1) it is Gauss's original notation and is therefore universally recognized and 2) it can be read in any text file whereas the L−shaped brackets above are special characters, which means that they are less transferable and less likely to be retrieved in the event the file is corrupted. The latter happened to one of my drafts. The concept of the greatest integer dates back to Ancient Greece. Using a straight edge one unit in length and a compass, the Euclidean Greeks could construct segments with lengths that were not integer multiples of the length of the straight edge. They would represent the length of these segments as the greatest integer multiple of the length of the straight edge and any remaining fraction of its length. Rather than represent a ratio as a fraction, the Euclidean Greeks would often express a ratio as an integer quotient with a remainder. For example, the Euclidean algorithm uses an iterative process of division with a remainder to compute the greatest common divisor of two numbers. The concept is simple enough, so why should you study it? While we know so much more about integers today than did the ancient Greeks, I believe our understanding of what integers are and how to work with them rigorously has improved little since ancient times. I argue that a narrow concept of the integer is not without consequence. I believe that a comprehensive study of the greatest integer function is justified for three main reasons. My first objective is to learn more about the properties and applications of integer functions. My second objective is to use integer functions to express more concepts and statements involving integers (such as in number theory, discrete math, and real analysis) mathematically so that we can work with them on a more solid foundation. My third objective is to expand coverage of number theory topics to the extent that this work can serve as a first−year course on the subject. The first reason is an end in itself. A lot of theoretical research is either a generalization or a spin−off of an earlier discipline. For example, the proof that the general quintic equation has no solution by radicals using the fact that its Galois group is not solvable led to a comprehensive study of solvable groups. Since number theory is the study of properties of integers and Gauss hailed it as the queen of mathematics, it seems only natural to expand our study to integer functions. In fact, Gauss’s third and most famous proof of quadratic reciprocity generated further interest in the study of sums involving the greatest integer function. Some notable contributions were made by Eisenstein, Hacks, Stern, and Zeller in the 19th century and by Berndt, Carlitz, and Dieter in the 20th century. The second reason is as practical as it is powerful. It would mean that the study of integer functions would not only unify several areas of mathematics, but it would also help us build a framework to study them in a more rigorous and straightforward manner. I am arguing that no amount of additional theory about integers can substitute for a mathematical expression. I believe that with sufficient study of the greatest integer function we can achieve this objective. First, since the greatest integer function is continuous in the sense that its domain includes all real numbers and discrete in the sense that its range is restricted to integers, the greatest integer function acts as a bridge between the continuous and the discrete by mapping any real number to its integer approximate not greater than itself. It follows that by studying the properties of integers and integer functions, we may be able to use integer functions to express, if not outright discover, relationships between constructs involving integers. Second, by expressing concepts and statements involving integers (including number−theoretic functions) in terms of the greatest integer function, we may be able to manipulate them mathematically using properties of integer functions. This will allow us to simplify, if not solve a greater variety of problems in number theory and construct more direct number theory proofs. For this reason, the study of the greatest integer function is indispensable to our understanding of integers and hence to number theory. Third, I already built up enough theory on the greatest integer function to use it for solving an increasing variety of problems involving integers. If I can get to this point working almost entirely alone, imagine how much further we can go with collaboration. The third reason necessarily follows from the first two. I am certain that if you enjoy number theory, then you will enjoy learning about the greatest integer function. My interest in the greatest integer function led me to study number theory because the study of integers is the most immediate application of integer functions. I spent years reading number theory texts looking for material on the greatest integer function and for new problems I could solve using my current knowledge of the greatest integer function. The vast majority of the material in my work on the greatest integer function is in discrete mathematics and number
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