You are here

An Experimental Introduction to Number Theory

Benjamin Hutz
American Mathematical Society
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts 31
[Reviewed by
Mark Hunacek
, on

Elementary number theory at the undergraduate level is, to some considerable extent, a study of patterns exhibited by the sets of integers. A lot of time during a typical course is spent studying questions like “What primes can be written as the sum of two squares?” or “What integers are congruent, modulo a given prime p, to a perfect square?”. These questions (and quite a few others) lend themselves nicely to experimentation and conjecture, and there are some books, such as Silverman’s A Friendly Introduction to Number Theory, that do a good job at stressing the importance of these activities. But the book now under review — written, coincidentally, by one of Silverman’s doctoral students — takes things to a level not generally found in competing textbooks.

This book is written with the expectation that its readers are willing to make a serious effort to collect data in response to specific questions, even to the point of using a computer, and will then be willing to think about drawing inferences and making (and proving) conjectures from this data. Algorithms are scattered throughout the book; they are written in ordinary English rather than any specific computer language. (Hutz points out in the preface that while familiarity with any one programming language is not a prerequisite for this book, a willingness to learn one is.) The author guides the reader through the process of making sense of this data and after a conjecture has been formulated, gives a precise statement and proof of the result. (Not all results are proved, of course. As is typical in number theory books at this level, some advanced results such as Dirichlet’s Theorem on primes in arithmetic progression are stated without proof.)

In addition to results that are stated and proved in the text, there are numerous guided projects (“Investigations”) that are described in detail. These start right at the very beginning of the book: after defining divisibility, for example, the author asks in a project for the reader to examine numbers according to the number of divisors that they have, and see if they notice any patterns. Other projects are more demanding and sometimes address topics that are the subject of research work.

The computational focus of this book is also reflected in the exercises, which come in three flavors. First, there are the ones labelled “computational exercises”, which require the student to do calculations. Some are simple enough to do with pencil and paper (and are marked with an icon to reflect that); others require the use of software. Second, there are the ones labelled “theoretical exercises”; these call for proofs or, on occasion, counterexamples. (The line between these first two categories is sometimes a bit blurry. I would describe, for example, a problem asking the reader to find all \(n\) such that \(\varphi(n)=n/2\) to be a theoretical exercise, but it’s listed here as a computational one.)

And finally, there are “Exploration Exercises”. The author describes these as “open-ended projects” that are intended to teach not only data-gathering, formulation of a conjecture, and construction of proofs, but also mathematical writing (“the student must describe the problem, their steps toward gathering data, and their conclusions in the form of conjectures.”). One of these Exploration Exercises, for example, has the reader explore the p-adic numbers; another investigates the representation of a positive integer as a sum of squares; yet another asks the reader to classify the algebraic integers in a quadratic number field. (As these examples illustrate, these Exploration Exercises can get very difficult.)

All three kinds of exercises come at the end of each chapter, and (wisely, I think) are not accompanied by solutions at the end of the book. There is also apparently no solutions manual available.

The book covers most or all of the standard topics in an elementary number theory course, but then also provides a selection of chapters allowing an instructor to choose between less standard ones. More specifically: the first five chapters cover topics that just about any book on this subject at this level will cover: divisibility and primes, congruences, quadratic reciprocity, arithmetic functions, and a chapter on applications to cryptography. Apart from the emphasis on computation, the discussions here are fairly standard; a student should find this material quite accessible.

The remaining chapters largely cover more special topics that do not appear in every book and will generally be covered based on the interests of the instructor: algebraic numbers, rational and irrational numbers (including a look at continued fractions), Diophantine equations (including a discussion of Pell’s equation and Waring’s problem), elliptic curves, and number-theoretic aspects of dynamical systems. A final chapter discusses polynomials, with an eye towards comparing and contrasting polynomial rings with the ring of integers. In a nice touch, Fermat’s Last Theorem for polynomials is proved; this doesn’t seem to be a topic that is discussed in a lot of number theory books.

I say “largely” in the previous paragraph because some of the material in these later chapters cover some topics that I personally think of as more common for a course like this. Even a simple and basic fact like the irrationality of \(\sqrt2\), for example, which is usually proved early on as a consequence of the fundamental theorem of arithmetic, is not proved until chapter 7, the chapter on rational and irrational numbers. (In the previous chapter, however, the author refers to \(\sqrt2\) as a “degree 2 algebraic number”, so he’s obviously assuming that the reader already knows this number is irrational.) Also, chapter 8, on Diophantine equations, covers some topics (e.g., Pythagorean triples and the case \(n= 4\) of Fermat’s Last Theorem) that are generally considered fairly standard fare for such a course.

The author states that the prerequisites for this text are a “basic understanding of functions from first semester calculus” and some knowledge of standard proof techniques. I generally agree with this, albeit with a few qualifications. For some chapters a prior exposure to algebra would be helpful. The author uses the term “ring” in chapters 9 and 11, for example, and in the chapter on algebraic and transcendental numbers, where algebraic number fields and their rings of integers are discussed, some prior exposure to rings and fields would also definitely be useful. The word “group” also gets used in chapter 9, but in a very non-essential way.

In addition, there were a couple of occasions when ideas from calculus beyond a basic understanding of functions were employed; Hensel’s lemma, for example, uses the derivative of a polynomial, and the statement of the Prime Number theorem uses (of course) the notion of a limit.

There were some times when I thought the author was guilty of slipping something by the reader without adequate discussion, and other times when I didn’t like a definition or the author’s choice of notation. For example, Hutz proves that the number \(\alpha=\sqrt{1+\sqrt3}\) is algebraic by showing, via the usual argument, that it is a root of the polynomial \(x^4-2x^2-4\); he then states, without explanation, that this polynomial is irreducible, and hence is the characteristic polynomial of \(\alpha\). One problem with this is that the term “irreducible polynomial” is not defined until about 130 pages later in the text. Another problem is that the irreducibility of this polynomial is not particularly obvious without knowing some tricks of the trade, but a student might think that it is from the way it is stated.

In addition, Hutz later states (in an Exploration Exercise) that \(\{1,\sqrt{d}\}\) is a \(\mathbb{Q}\)-basis for the algebraic number field \(\mathbb{Q}(\sqrt{d})\)but the reason he gives for this statement is that every element of the field can be written in the form \(a+b\sqrt{d}\). He does not mention linear independence or the fact that every element can be written uniquely in that form. Then, later on in the same exercise, we read \[\mathbb{Q}(\sqrt2,\sqrt3)=\{a+b\sqrt2+c\sqrt3\},\] a statement that should make any person who has taught elementary field theory cringe.

I also noticed one omission that struck me as a missed opportunity. Hutz mentions Fermat’s Last Theorem (FLT) on occasion, and later provides a chapter introducing elliptic curves — yet does not mention in this chapter that elliptic curves played a major role in the eventual resolution of FLT. Obviously a detailed description is beyond the scope of any undergraduate textbook, but Silverman, for example, ends his book with a two-page overview of the connection; at least the reader learns that elliptic curves have something to do with FLT, even if they don’t quite know what. See also Kraft and Washington, An Introduction to Number Theory with Cryptography, for a short chapter on these connections.

Issues like these are relatively few and far between, however, and by and large the book is an enjoyable one. The author writes clearly, and there are lots of examples. The big issue for most people, I think, will be the extent to which they want to emphasize computation in the course.

When an author writes a book with a distinctive point of view, the value of that book to an instructor obviously depends on the extent to which he or she buys into that viewpoint. Because this text does go beyond calculation and provides statements and proofs of the major results of elementary number theory, it might be possible to teach a course based on this book and never have a student touch a computer: simply take the data offered in the text on faith, don’t assign many of the computational exercises, etc. But why bother? If you’re not interested in emphasizing calculation and experimentation, there are lots of other books on the subject from which to choose. If, however, you see the value of stressing calculation and computers in a first course in number theory, then this book is one that you will want to take a good look at the next time you teach number theory.


Buy Now

Mark Hunacek ( teaches mathematics at Iowa State University.

See the table of contents in the publisher's webpage.