A Pythagorean Introduction to Number Theory

This book offers an interesting variation on the traditional undergraduate number theory course. It is organized around the general theme of number-theoretic questions relating to the study of right triangles. Number theory arises here, of course, because the familiar Pythagorean theorem for such triangles is an example of a Diophantine equation, the study of the integral or rational roots of which can lead to interesting questions. For example, the classification of primitive Pythagorean triples is a topic that just about every undergraduate number theory text covers, as is the question of which positive integers can be written as the sum of two squares. There are other questions that can be asked, however, and their study leads to some interesting, yet relatively accessible, mathematics. 

 

The book is divided into two parts of roughly equal length. The first part, consisting of the first seven chapters of the book, should be entirely accessible to undergraduate students taking an introductory course in number theory, although some material here (such as the discussion of congruent numbers in chapter 4) is not generally covered in such a course. The second part, consisting of the remaining seven chapters, is somewhat more sophisticated; while some of the material here could be covered in an introductory course, a substantial portion of it would be better introduced in a second course. 

 

In more detail: the book begins with a chapter offering several proofs of the Pythagorean theorem. (The idea of proving something in multiple ways is one that comes up again in the text, and is, I think, a very attractive feature of the book.) Proofs of the Pythagorean theorem typically come up, of course, in a text on geometry rather than number theory, but the discussion here seems entirely appropriate given the central role that that theorem will play in the ensuing discussion. And anyway, the Pythagorean theorem is something that all undergraduate mathematics majors should, but often do not see proved.

 

Chapter 1, like all other chapters in the book, ends with a section titled “Notes”. These sections discuss historical issues (paying more than the usual amount of attention to non-European aspects of mathematics history), survey other issues that are relevant to the subject matter of the chapter, and provide bibliographic references. These end-of-chapter sections are another very attractive feature of the book; they are nicely written and quite informative.

 

Chapter 2 discusses the basic topics of number theory: divisibility, primes, congruences, Fermat’s Little Theorem, and primitive roots (including a classification, with proof, of the precise values of \(n\) for which a primitive root mod \(n\) exists). The exposition here, while clear, is quite efficient: this material can easily take up a third of a typical undergraduate number theory semester, yet is done here in about 40 pages. Rather surprisingly, the author does not prove in this chapter that there are infinitely many prime integers, but this will be proved later in the book. (I would have preferred to see it proved earlier, but obviously this is not a serious issue; nothing prevents an instructor from proving this result earlier rather than later.) 

 

Basic divisibility arguments are then used to characterize, in chapter 3, primitive Pythagorean triples. Following this, the standard geometric argument (using chords of the unit circle connecting \((-1,0)\) to other rational points on the circle) is given to produce an alternative proof. The geometric argument is used to analyze other equations, and finally, all of this material is used to prove Fermat’s Last Theorem for exponent 4. The Notes section of this chapter is a particularly interesting one that discusses Fermat’s Last Theorem a little more and also introduces the student to elliptic curves and the abc conjecture.

 

The next chapter looks at a different aspect of right triangles, their area, and asks the question: what numbers are the area of a right triangle with integer-length sides? This quickly leads to the concept of congruent number (i.e., a number that is the area of a right triangle with rational sides); there is currently no known precise characterization of precisely what integers are congruent numbers, but the author proves some results along these lines. 

 

Chapter 5 addresses the question of what natural numbers can appear as the length of a side of a right triangle. With regard to the hypotenuse, this question quickly devolves into the question of what natural numbers can be written as the sum of two squares. This is a familiar topic in elementary number theory courses but the author’s approach here is interesting: he establishes this result as a consequence of facts about the Gaussian integers, which are introduced from scratch. Basic facts about the ring of Gaussian integers (the existence of a Euclidean algorithm and unique factorization, the characterization of irreducible elements, etc.) are fully worked out. 

 

The question of sums of squares leads naturally to the idea of primes that are congruent to \(1 \mod 4\), and these primes motivate the material of the next two chapters. After (finally!) proving the existence of infinitely many primes, the author also proves that there are infinitely many primes of the form \(4k + 1\) and \(4k – 1\). (Dirichlet’s theorem on primes in arithmetic progression had been previously mentioned in the Notes section of the previous chapter.) The study of primes of the form \(4k + 1\) leads to the study of quadratic residues, and in turn leads to the law of quadratic reciprocity, which is stated in chapter 6 and proved (using Gauss sums and some basic facts about algebraic integers, developed in an Appendix) in chapter 7. 

 

This takes us to Part II (“Advanced Topics”) of the book. The author begins by studying the number of solutions of the Pythagorean equation modulo \(n\), an enterprise that leads to Hensel’s theorem, the proof of which is an exercise. Then the question of sums of squares (discussed earlier for two squares) resurfaces, this time for two, three and four squares. The theorems on these topics are first proved using geometric lattice-point arguments, and then, in the next chapter, the theorem on sums of four squares is given a different proof, using quaternions. Sums of three squares, via the theory of quadratic forms, are discussed in chapter 12.

 

The last two chapters of the text are the most difficult, and least self-contained. In chapter 13 the author proves an asymptotic formula (first established by Lehmer in 1900) for the number of primitive Pythagorean theorems with bounded hypotenuse. In chapter 14, we return to a topic from chapter 3, rational points on the unit circle; the main result of this chapter is a very difficult theorem concerning the distribution of these points according to their height, a concept introduced in the chapter.

 

These 14 chapters are then followed by three appendices. The first is quite elementary and provides some background material on things like the Pigeonhole Principle (with Dirichlet’s theorem on approximation of irrational numbers by rational ones given as a nice example); the second covers the basics of algebraic numbers and integers; the third discusses the free mathematical software SageMath.

 

The book is quite nicely written, with good motivation and a substantial supply of examples. As noted earlier, it also benefits from the author’s attention to the historical aspects of the subject and his penchant for giving alternative proofs of various results. There are a decent number of exercises, both computational and proof-based, covering a broad spectrum of difficulty. The author states that hints to some of the more difficult ones will appear on the book’s website, but apparently none have, as of this writing, yet made it up there.

 

Prerequisites for the text are (for the first part, anyway) not unduly burdensome. No prior background in abstract algebra is necessary, though on occasion the author mentions a word like “ring” without definition. No real background in ring theory is necessary to understand the book, however. (Since the author does feel free to drop the occasional algebraic phrase, I’m surprised that he didn’t make the obvious connection between some of the topics covered in chapter 2 and basic group theory.) Part 2 of the book does require considerably more mathematical sophistication, and, as previously noted, in several chapters a background in analysis is required.

 

I noticed few misprints or typos, and the ones that I did see are minor: for example, in theorem 4.4 on page 83, the author uses the ordinary letter “S” to denote the set of congruent numbers, but had previously used a stylized script version of that letter in the definition.

 

And speaking of the letter S, I do think that the index of the book could stand some improvement. Anyone looking to see where Andrew Wiles is mentioned in the text will not find that answer under “W”; instead, the reference to him appears under “S”, for “Sir Andrew Wiles”. The word “prime” does not appear in the Index, which seems a curious omission in a textbook on number theory. And I was unable to find any reference to the Law of Quadratic Reciprocity under the letters “L”, “Q” or “R”; I finally got lucky and found it under “E” (for “Euler”).

 

As a text for a course, I see only one possible drawback, and that one is obvious and unavoidable: the author’s approach to the subject, while novel and interesting, has the inevitable consequence of resulting in the omission of certain topics that an instructor might want to cover in an introductory course. Continued fractions, for example, are not discussed in the text, and, perhaps of more concern for most instructors, neither is cryptography, which seems, more and more, to be a topic that people want to see covered in an undergraduate number theory course. Even if not considered as a text for such a course, however, the book has several other potential uses: it could be used as a text for a second semester course in number theory or “special topics” course, or as a text for an introductory graduate course. It’s also just an interesting book to have on one’s shelf.

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.