Number theory is a branch of mathematics that deals with the properties and relationships of integers. It is one of the oldest and most fascinating areas of mathematics, with many unsolved problems and intriguing conjectures. Number theory also has many applications in cryptography, coding theory, combinatorics, and other fields.
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In this article, we will give a brief overview of a book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. The book is called Number Theory: A Lively Introduction with Proofs, Applications, and Stories, and it is written by James Pommersheim, Tim Marks, and Erica Flapan. It was published by Wiley in 2010.
The book covers the main topics of elementary number theory, such as divisibility, primes, congruences, modular arithmetic, quadratic residues, Diophantine equations, cryptography, and more. It also introduces some advanced topics, such as elliptic curves, Fermat's Last Theorem, and the Riemann Hypothesis. The book aims to present the concepts and theorems of number theory in a lively and engaging way, using proofs, applications, and stories.
The book emphasizes the importance of proofs in mathematics and teaches the students how to write clear and correct proofs using various methods and techniques. The book also provides many examples of proofs and exercises for practice. Some of the proofs are preceded by numerical proof previews, which are numerical examples that help the students understand the statements and ideas behind the proofs before they are formalized in more abstract terms.
The book shows how number theory can be applied to various areas of mathematics and science, such as cryptography, coding theory, combinatorics, geometry, algebra, physics, and biology. The book also explains how number theory can be used to solve real-world problems, such as encryption, error detection and correction, digital signatures, Sudoku puzzles, magic squares, RSA algorithm, Diffie-Hellman key exchange protocol, ElGamal encryption scheme, elliptic curve cryptography, primality testing algorithms, factorization algorithms, and more.
The book includes many stories that illustrate the history and development of number theory and its connections to other fields of knowledge. The book also introduces some of the famous mathematicians who contributed to number theory, such as Pythagoras, Euclid, Fermat, Euler, Gauss, Legendre, Lagrange, Riemann, Hardy, Ramanujan, Wiles, and others. The book also features some math myths, which are fictional stories that introduce an important number theory topic in a friendly and inviting manner.
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The book is intended for undergraduate students who have some background in calculus and linear algebra. It can be used as a textbook for a one-semester or two-semester course on elementary number theory or as a supplementary text for courses on discrete mathematics or cryptography. The book can also be useful for graduate students who want to review or learn some basic concepts and techniques of number theory. Moreover, the book can be enjoyed by anyone who is interested in learning more about number theory and its applications.
The book can be purchased from various online platforms or from local bookstores. The ISBN of the hardcover edition is 978-0-470-42413-1[^1^]. The book also has a companion website[^2^] that provides additional resources for instructors and students, such as solutions to selected exercises, slides, worksheets, projects, and links to other websites related to number theory. 8cf37b1e13