Preface

Preface

The following text is an extensive update of an original manuscript by Professor W. Edwin Clark (now Emeritus) of the University of South Florida, written in 2002 and made freely available on his website. Professor Clark’s gracious use of what he termed “copyleft” status for his book, wherein I and other instructors have been able to distribute his text freely to our students and edit it to suit our needs, has been a great boon to many.

As I have taught out of Professor Clark’s text for multiple semesters, I have made several changes in my own presentation of the material, and because of Professor Clark’s generous permission, I have undertaken to adapt the book to reflect these changes.

The text is designed for a one-semester course and for self study. As Clark did, I have assumed that students have some familiarity with basic set theory and high school-level algebra (including imaginary numbers, if the chapter on Gaussian integers is to be taught), as well as the notion of a limit in one chapter. As Clark wrote in his original preface,

The text requires only a certain amount of mathematical maturity. And, hopefully, the student’s level of mathematical maturity will increase as the course progresses.

I have tried to keep the same foccus.

Clark’s book was largely self-contained, and I have tried to maintain that quality as I have made changes and added material to the text. Following is a brief description of most of the changes made to the text.

• Textual changes to improve flow were added to various chapters.
• I have moved parts of the original book’s preface to various locations in text.
• An introductory chapter (which appears as Chapter 1) was added to the text to engage student interest and convey the excitement of experimentation in number theory.
• Some exercises were reordered, and the wording in and around various exercises was changed to correct grammatical mistakes and/or make the intent clearer. Exercises were collected at the ends of the chapters, with additional exercises included, in several chapters of the text.
• Major changes were made to what is now Chapter in an attempt to motivate and improve the presentation on mathematical induction.
• Material on the floor and ceiling functions has been moved to the initial chapter discussing the integers, where its relation to the Well-Ordering Principle is mentioned. I have substituted an alternate proof of the Division Algorithm in Chapter and reduced the use of the floor function in that chapter to a mention and an exercise.
• The chapter on base \(b\) representations of \(n\) has been moved earlier in the text (it is now Chapter ) to appear just after the chapter on the Division Algorithm. In this way students see two distinct uses of the Division Algorithm in the chapters that follow (the Euclidean Algorithm being the other).
• The concept of the least common multiple was introduced along with the greatest common denominator and developed through a few chapters’ exercises.
• The chapter on Bezout’s Lemma (currently Chapter ) was supplemented with additional motivation and exercises.
• The chapter on Blankinship’s Method (currently Chapter ) was retitled and now includes a brief description of the Extended Euclidean Algorithm; exercises were modified to adopt this change.
• As a means of commenting on prime factorizations and providing some context for the sums-of-squares discussion at the end of the text, a new chapter is devoted to the Gaussian integers.
• The chapter on Fermat and Mersenne primes has a broader theme, with a new introduction about functions that generate prime numbers (including a mention of the Green–Tao Theorem on arithmetic progressions in the primes).
• Information on the prime counting function and Mersenne primes found has been updated to include the latest records as of July 2021.
• The material on Euler’s totient function has been expanded upon and moved to the chapter with other number theoretic functions.
• The introduction of perfect numbers was moved from the chapter on number theoretic functions to Chapter .
• For the sake of pacing, I have compressed three chapters from Clark’s original text into one (currently Chapter ), omitting the notion of a complete system of residues, as it was not needed anywhere else in the text.
• Clark’s original text had the virtue of distinguishing between \(\mathbb{Z}_m\) and the sets \(\{0,1,\dots,m-1\}\) through the use of the notation \(J_m\), \(\oplus\), and \(\odot\). However, in light of student difficulties due to competing notation used in other mathematics classes (notably abstract algebra classes), and because the formalism is not built upon very much in the chapters that follow, I have blurred the lines between these two rings by renaming Clark’s \(J_m\) as \(Z_m\) (without the blackboard bold font) and not circling the symbols of the associated binary operations.
• A new chapter discusses the Chinese Remainder Theorem.
• A discussion of Wilson’s Theorem was added to Chapters and , and content in Chapter was reordered.
• The chapter on RSA (Chapter ) has been reorganized, with the addition of supporting examples and text and several exercises.
• A new chapter discusses representation of integers by sums of squares, connecting the end of the course with Chapter 1 and pointing interested students towards results on quadratic residues.
• References to Clark’s supplementary worksheets in Maple (eg., in Chapter ) have been removed. I recommend that instructors supplement the text with computational experiments using technology that best suits their students’ needs.

My goals for students are identified as “three wishes” in Chapter 1. I am happy to hear of feedback on the text (including the correction of errors); I may be contacted via email at barrus@uri.edu. In accordance with the terms of the copyleft status Clark originally imposed, this text, including my changes, may be used under the Creative Commons BY-NC-SA 4.0 license. As such it may be freely modified and distributed with attribution (please mention both W. Edwin Clark and myself), and later adapters are expected to license their editions the same permissions they have found with this one.

Michael D. Barrus

University of Rhode Island

August 2021