Elementary Number Theory (Barrus and Clark)

The themes

The questions and application above illustrate some themes we’ll run into throughout the text. We will look at various forms in which integers can sometimes be written. We will look multiple times at producing or identifying prime numbers. We will look for integer solutions to equations, and we will use properties of the integers to design algorithms to accomplish many different tasks. As you skim the table of contents now, and perhaps return to this chapter occasionally as you go through the text, you’ll see these themes (and others) played out again and again.

There’s another happy feature of number theory: many of the results we’ll discuss won’t be necessarily hard to recognize when you see them in action—in fact, several of the results will pop up quite easily as we look for patterns among multiple examples. (Of course, as mathematicians we’re never satisfied until we can rigorously justify our observations through proof, but I hope you’ll find the proofs in this text pleasant to digest, as well.)

Because number theory is about patterns in the integers, you will be well served to work out several numerical examples of ideas you encounter in the text and exercises. If you are familiar with writing simple programs in a computer algebra system (eg., CoCalc, Maple, Mathematica, MATLAB) or in a programming language, please try often to turn what you see in your studies into programs. You will be able to see many more examples this way, and the thought processes involved in writing your programs will enhance your understanding of number theory. As contemporary number theorist William Stein has said,

A computer is to a number theorist, like a telescope is to an astronomer. It would be a shame to teach an astronomy class without touching a telescope; likewise, it would be a shame to teach this class without telling you how to look at the integers through the lens of a computer.

Because there is a wide variety in the programming/computing environments with which students may be familiar, this book will not focus on any one computational system; however, you are heartily encouraged to pick one and dig deeply into number theory with it.

Three wishes

This book will have succeeded if it helps you do the following (not necessarily in order of importance):

• appreciate the beauty of patterns found in the integers;
• appreciate some of the practical applications of number theory;
• continue your growth in mathematical maturity and skill.

Here’s to our success…Let’s get started!