An Illustrated Theory of Numbers

Visualization is a very comfortable way to contemplate on scientific facts. For mathematical facts, visualization includes:

• Proofs based on a picture (as in many proofs for elementary geometry),
• Pictures describing a proof (several facts in calculus),
• Pictures, charts, diagrams, and even multimedia productions that help us see a result.

All of the above help the reader to have a better understanding of the result under study, and also help to see possible patterns. While the “Theory of Numbers” is known for its abstract face, the book under review gives illustrated versions of many number-theoretic concepts. There are many pretty color figures giving conceptual pictures and diagrams, visualized computations, representations of the key insight in proofs, as well as several tables.

The first page of Chapter 1 shows the picture of an ancient copy of Euclid’s Elements, Book VII, Proposition 1, which shows some elementary diagrams like the ones presented in this chapter. This may represent, I believe truly, that the book under review is a modern rewriting of parts of the Elements, plus of course some newer topics.

The book consists of three parts plus the introductory Chapter 0, in which the author focuses on the basic ideas of seeing arithmetic, following geometric ideas of summing integers, divisibility and divisors, and the division algorithm.

Part I (Foundations) includes Chapters 1–4. In Chapter 1, the author studies the Euclidean Algorithm and the concepts of greatest common divisor and least common multiple. Chapter 2 is about prime numbers, and starts with prime factorizations as tree diagrams. Within some few pages, the author introduces the amazing facts and conjectures on the distribution of primes, mentioning the Green-Tao Theorem, the Riemann Hypothesis, and PolyMath’s recent result on the gaps in the distribution of primes, with some very nice pictures and diagrams. The author comes back to elementary topics related to primes, and meanwhile introduces the functions giving number and the sum of divisors, leading to the definition of perfect numbers. In Chapter 3, the author studies constructible numbers, rational and irrational numbers, and gives a picture proof of the Dirichlet approximation theorem via Ford circles. Chapter 4 is about Gaussian and Eisenstein integers. The author provides a picture study of Gauss’s circle problem and its analogue for Eisenstein integers.

Part II (Modular Arithmetic) comprises the next four chapters. Foundations of modular arithmetic such as the definitions, some divisibility tests, linear congruences, and multiplicative inverse are contained in Chapter 5. This chapter also covers basics of modular arithmetic of polynomials and the prime number theorem for polynomials modulo a prime. Since multiplication is repeated addition and exponentiation is repeated multiplication, to understand these important operations in modular arithmetic the author studies modular dynamics in Chapter 6, including the Fermat-Euler Theorem, the concepts of order and primitive root, and an application in cryptography. In Chapter 7 the author studies the Chinese Remainder Theorem and states more properties of the Euler function with some applications on the public-key cryptosystem RSA. Chapter 8 follows Zolotarev’s proof of quadratic reciprocity law, which fits well with the theme of dynamics of modular arithmetic. Meanwhile the author studies Wilson’s Theorem, Fermat’s Christmas Theorem, Minkowski’s theorem in the plane, and Legendre’s symbol.

Part III (Quadratic Forms) consists of Chapters 9–11, where details of a visual approach known as the “topograph,” due to John H. Conway, is used. (The main reference for this is Conway’s The Sensual (Quadratic) Form.) This method is used to solve quadratic Diophantine equations of two variables and to visualize those solutions. In Chapter 9, the author presents a two-dimensional Hop and Skip problem, and introduces the domain topograph and some related concepts. Chapter 10 starts with some practical examples of Conway’s method, including a proof of Fermat’s Christmas Theorem and some analogues. The author studies the discriminant of a quadratic form, then considers the important concept of class number and states the class number 1 theorem of Heegner, Stark and Baker.

In Chapter 11, the author studies indefinite quadratic forms and, as an application, considers solution of Pell’s equation. In this chapter, computations that visualize Gauss class number 1 conjecture, and also the Markov invariant are considered. The book ends with indexes of Theorems, Terms, and Names. There is a bibliography that includes a list of references giving several ideas to make number theoretic concepts visual.

Each chapter starts with a picture or diagram indicating what is going on the chapter, and ends in some historical notes and a number of exercises. The book includes most of the topics in a typical elementary course in number theory, insisting on the concepts related to quadratic equations and quadratic forms. I believe that this book is a very interesting text for a first undergraduate course. Graduate students and researchers will also find some good ideas here, particularly on how develop a concept with pictures and diagrams. If the instructor doesn’t choose this book as the text, I suggest it strongly as a supplement for students, allowing them to follow their number theory course with some visualization in parallel.

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Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.