Review of Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, and Erica Flapan
Number theory is the branch of mathematics that studies the properties of natural numbers and their relationships with other mathematical objects. It is one of the oldest and most fascinating areas of mathematics, with applications ranging from cryptography to music theory. However, learning number theory can be challenging for students who are not familiar with its concepts and techniques.
That is why a book like Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, and Erica Flapan is a welcome addition to the literature on this subject. This book aims to provide a rigorous yet accessible introduction to elementary number theory along with relevant applications. It covers topics such as divisibility, prime numbers, modular arithmetic, congruences, Diophantine equations, cryptography, quadratic reciprocity, continued fractions, and more.
What makes this book different from other textbooks on number theory is its engaging style and pedagogical features. The authors use readable discussions to motivate new concepts and theorems before their formal definitions and statements are presented. They also use numerical examples to illustrate the statements of the theorems and the ideas behind their proofs, before the proofs are formalized in more abstract terms. This helps students to develop a concrete understanding of number theory and its methods.
In addition, the book includes many applications of number theory that are explained in detail throughout the text. Some of these applications are well-known, such as encryption schemes, digital signatures, and error-correcting codes. Others are less common, such as magic squares, Sudoku puzzles, musical scales, and card tricks. These applications show how number theory can be used to solve real-world problems and to create interesting patterns and phenomena.
A unique feature of the book is that every chapter includes a math myth, a fictional story that introduces an important number theory topic in a friendly and inviting manner. For example, one math myth tells the story of how Gauss discovered a formula for the sum of the first n natural numbers when he was a child. Another math myth tells the story of how Fermat's Last Theorem was finally proved after centuries of attempts. These stories help to spark students' curiosity and interest in number theory.
The book also contains many exercises that challenge students to apply their knowledge and skills. Some of these exercises are in-depth explorations that develop a topic that is related to the material in the section. For example, one exploration asks students to investigate the properties of Fibonacci numbers and their connections with other mathematical concepts. Another exploration asks students to construct their own public-key cryptosystem using modular arithmetic. These exercises allow students to deepen their understanding and appreciation of number theory.
Overall, Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, and Erica Flapan is an excellent book for anyone who wants to learn more about this fascinating subject. It combines rigor and clarity with fun and creativity, making it suitable for both beginners and advanced learners. It is also a valuable resource for teachers who want to inspire their students with the beauty and power of number theory.