*Introduction to Proof Through Number Theory* is a fun book that introduces students to mathematical thinking and proof writing through a deeper understanding of elementary concepts in Number Theory. The book’s informal style with humor, games, puzzles, and connections to popular culture with references to music, art, movies and literature lightens up the material without compromising rigor.

This book covers an extensive list of topics in great depth. Owing to the variety of topics instructors can choose which examples, sections and chapters to focus on in a one semester course. Instructors have the choice to do as little or as much in each topic. The topics and sections within each Chapter flow well. Each chapter in the book begins with explicit goals to highlight the important content in the chapter. Many theorems are proved formally as well as visually to enhance understanding. For instance, the visual proof of \( \sum_{i=1}^{n}(2i-1) = n^{2} \)and in addition to the proof by induction makes the formula more relatable. A key feature of this book is that exercises are interspersed along with the chapter or section material rather than at the end of the section or chapter. Students can practice the exercise(s) after understanding the proofs of theorems or solved examples leading up to the exercise(s). This way of chunking the material could be helpful for students. Many of the exercises are straightforward and can be assigned for all students as homework. The ideas for solving the exercises lie in the proofs of theorems or solved examples. For many exercises, the hints provided at the end of each chapter for the exercises are also helpful.

Chapter 1 introduces students to proofs by deduction, contradiction and contrapositive by proving some basic facts about integers, composite numbers and prime numbers. From the very beginning the style of the book is that on occasion results from a later chapter or section are used in the proof of earlier results to maintain the flow of the topic. This may help the reader to see connections across chapters.

An enthusiastic treatment of proofs by induction is given in Chapter 2. The statement of the first principle of mathematical induction is explained in many ways: visually with dominoes, a dialogue between people, and formal easy to understand examples. Students can follow the first examples and try similar exercises right away before moving onto more challenging examples and exercises or moving on to strong induction. There is also an entire section on proving Fibonacci identities using induction which is fun. A more advanced student can benefit from examples that relate to calculus such as comparing exponentials to polynomials or exploring sums of powers of integers and their connection to integration. The principle of strong induction is also explained visually with dominoes and is used to prove the existence part of the prime factorization theorem.

Chapter 3 on logic begins with a detailed solution of a logic puzzle followed by a thorough study of the rules of logic using truth tables and colorful Venn diagrams to visualize the rules. Detailed examples of proofs by contrapositive and contradiction are given including the proof of why the square root of 2 is irrational and connections are made to proofs in previous chapters that use these techniques. This example is generalized to a theorem characterizing when the square root of an arbitrary positive integer n is irrational. To make the proof of the generalization understandable, a few more results such as \( \sqrt{p} \) is irrational if \( p \) is a prime and \( \sqrt{pq} \) is irrational if \( p \) and \( q \) are prime, are proved prior to the general result. This illustrates the depth and rigor of exposition in this book. Chapter 3 ends with proving or disproving existential and universal statements written using symbols rather than the spoken language.

Chapters 1, 4, and 6 include topics that are covered at the beginning of an elementary number theory course. Concepts such as gcd, well ordering principle, division and Euclidean algorithms, the fundamental theorem of arithmetic, lcm, and properties of congruence mod andnand are covered in detail in these chapters. In Chapter 5 on sets and functions, there are many exercises to especially enhance understanding of “is an element of” and “is a subset of” by proving or disproving statements involving the empty set which is interesting. The basic set operations, introduction to functions, and injective and bijective functions, and cardinality of finite sets are covered well with visuals to support the examples or proofs of theorems. A discussion of infinite sets and Cantor’s diagonal argument is not included in this book.

A few topics from Chapters 7 and 8 such as the addition and multiplication rules for cardinalities of finite disjoint unions and cartesian products, respectively, and the notion of equivalence classes are typically covered in an introduction to proofs course.

Since the general style is informal, and concepts are explained in different ways, and exercises and solved problems range from easy to challenging, this book is suitable as a textbook or a reference book for an introduction to proofs course. Moreover, this book can serve as a textbook for a two-semester sequence: Introduction to proofs course followed by a Number Theory course. Topics such as Linear Diophantine Equations, Congruence Equations, Fermat’s Little Theorem, Euler’s Theorem, primitive roots, and quadratic residues can be covered in the second semester. This book can also be used as a standalone textbook for an introduction to number theory course.

Hema Gopalakrishnan is an associate professor of mathematics at Sacred Heart University.