Explorations in Number Theory

Anne Rice wrote in the foreword to a collection of Kafka short stories, “Don’t bend; don’t water it down; don’t try to make it logical; don’t edit your own soul according to the fashion. Rather, follow your most intense obsessions mercilessly. Only if you do that can you hope to make the reader feel every particle of what you, the writer, have known and feel compelled to share.” The authors of Explorations in Number Theory capture the spirit of this writing approach by flouting traditional textbook writing. Rather, they write an introductory textbook to number theory that they would have wanted as an undergraduate—for better or worse. 

 

Chapter 1 explores the concept of what type of objects could be considered ‘numbers.’ Chapter 2 reflects on historical aspects of number theory and focuses on elliptic curves and Fermat’s Last Theorem, the Pythagorean Theorem, and Diophantine equations. Chapter 3 addresses many topics: integral domains, divisors, GCD/LCM, linear Diophantine equations, the Euclidean algorithm, Fundamental Theorem of Arithmetic, valuations, etc. Chapter 4 parallels Chapter 3 but in the world of modular arithmetic. This chapter also covers much of the text’s abstract algebra in addition to the totient function, the Chinese Remainder Theorem, etc. Chapter 5 continues the themes of Chapter 3 and 4 but with the Gaussian integers. Chapter 6 discusses more general ring topics, e.g. algebraic integers, norms, Euclidean domains, quadratic fields, etc. Chapter 7 covers quadratic residues and quadratic reciprocity. Chapter 8 addresses all the traditional p-adic topics: valuations, absolute values, \(\mathbb Z_p \) and \( \mathbb Q_p \), Hensel’s Lemma, etc. Finally, Chapter 9 briefly addresses additional topics such as Fermat’s Last Theorem, Lagrange’s Four-Square Theorem, elliptic curves, cryptography, and factorization of ideals. 

 

As the authors state, the book does not try to be comprehensive and self-contained but rather be modern and flexible. Primarily focusing on Diophantine equations and the concept of ‘numbers,’ the overarching goal of the book being to generally spark an interest in number theory through these topics. The focus is strict as the book has little to say about any topics from analytic, additive/multiplicative number theory, etc. Although, this need not be a downside. The book assumes some familiarity with vector spaces (though this seems not to be critical) and some ‘maturity with proofs.’ However, it does not assume exposure to abstract algebra and develops these topics as necessary. There are many similar, wonderfully written introductory number theory texts with similar content and focus, e.g. Stillwell’s Elements of Number Theory, Hutz’s An Experimental Introduction to Number Theory, and especially Lozano-Robledo’s Number Theory and Geometry. What differentiates this book is two things: the Explorations and tone. 

 

The book integrates inquiry-based learning problems throughout the text called ‘Explorations.’ These are generally good quality exercises to help students engage with topics; however, several do feel more like additional, extended exercises than inquiry-based problems. Broadly speaking, the exercises in the book are of good quality and present in sufficient quantity. Although, there are problems that some students may find more difficult because they do not necessarily immediately resemble the content of the section or chapter—which can be a good thing! In support of the book’s goals, there are also ‘general number theory awareness’ exercises to help build student’s awareness of and interest in number theory topics. These are generally good, open-ended problems for students to explore topics and build connections; however, some of these problems can be open-ended to the point of vagueness. The programming exercises (often in Sage) are less useful and often require more programming skills than some students would have. There are not many of these exercises and they are not a core component of the text. It would have been wonderful to see these more present and integrated into the Explorations as the basis for inquiry. Generally, instructors may find using the exercises difficult depending on the course structure because there are problems on the same or similar topics scattered throughout the book. While there are gems to be found, this may often lead to the instructor or students having to skip about the text. This even occurs with core book concepts. For example, the various elliptic curve topics are found in Chapter 2, 4–7, and 9. However, in the spirit of inspiring interest and reinforcing concepts, these revisitations are a definite positive. 

 

The truly unique thing about the book is the overall tone of the text. From the literal first sentence, the authors try talking to students rather than at them. The tone is casual—almost conversational— and filled with humor and wit that is almost never found in a textbook. There are definitely well- written sections with informative visualizations, e.g. Chapter 5 on the Gaussian numbers. Indeed, Chapter 8 on the p-adics is a particularly beautiful exposition for undergraduates. It can be terse, especially in Chapter 9 or the introductory abstract algebra sections. But the abstract algebra background is understandable and even reminiscent of Gallian’s Contemporary Abstract Algebra. However, the casual nature can be a two-edged sword. The reader’s experience can vary based on if they find the ever-present humor and tone enjoyable or distracting—particularly the jumps to footnotes for jokes. At times, it might be difficult to tell what is a joke and what is a colorful, colloquial description of the underpinning mathematics. Most of all, the book is seemingly ordered from the perspective of conversing with students and not structuring a course. There are many instances across the book of ordering that instructors may find frustrating. For example, Euclid’s lemma occurs after GCD/LCM and the Fundamental Theorem of Arithmetic. This is then followed by various topics, including the definition of the p-adic valuation, before finally seeing Euclid’s Theorem. The ‘abstract’ is often presented before the ‘concrete’—a personal choice. 

 

Explorations in Number Theory certainly does not try to be all things for all people. Instructors, students, or readers who appreciate its casual tone, ‘conversation based’ ordering, themes, style, etc. will find it thrilling from start to finish and be happy to read it or use it for a course. Others will appreciate it for what it is and politely move on—with likely less in the middle when presented with alternative texts. This will be especially true for instructors with fixed course structures in mind and cannot understand how to merge their goals and ideas with the book’s ordering and style.

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Dr. Caleb McWhorter is an Assistant Professor of Mathematics at St. Thomas Aquinas College. His research is in number theory, primarily in arithmetic geometry focused on elliptic curves.