Higher Arithmetic: An Algorithmic Introduction to Number Theory (Student Mathematical Library, 45)

Although number theorists have sometimes shunned and even disparaged computation in the past, today's applications of number theory to cryptography and computer security demand vast arithmetical computations. These demands have shifted the focus of studies in number theory and have changed attitudes toward computation itself. The important new applications have attracted a great many students to number theory, but the best reason for studying the subject remains what it was when Gauss published his classic Disquisitiones Arithmeticae in 1801: Number theory is the equal of Euclidean geometry--some would say it is superior to Euclidean geometry--as a model of pure, logical, deductive thinking. An arithmetical computation, after all, is the purest form of deductive argument. Higher Arithmetic explains number theory in a way that gives deductive reasoning, including algorithms and computations, the central role. Hands-on experience with the application of algorithms to computational examples enables students to master the fundamental ideas of basic number theory. This is a worthwhile goal for any student of mathematics and an essential one for students interested in the modern applications of number theory. Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990), Linear Algebra (1995), and Essays in Constructive Mathematics (2005). For his masterly mathematical exposition he was awarded a Steele Prize as well as a Whiteman Prize by the American Mathematical Society. Readership: Undergraduates, graduate students, and research mathematicians interested in number theory. Cover S Title Higher Arithmetic.. An AlgorithmicIntroduction to Number Theory Copyright © 2008 by the American Mathematical Society ISBN 978-0-8218-4439-7 QA241 .E39 2008 512.7—dc22 LCCN 2007060578 Contents Preface Chapter 1. Numbers Exercises for Chapter 1 Chapter 2. The Problem [omitted] Exercises for Chapter 2 Chapter 3. Congruences Exercises for Chapter 3 Chapter 4. Double Congruences and the Euclidean Algorithm Exercises for Chapter 4 Chapter 5. The Augmented Euclidean Algorithm Exercises for Chapter 5 Chapter 6. Simultaneous Congruences Exercises for Chapter 6 Chapter 7. The Fundamental Theorem of Arithmetic Exercises for Chapter 7 Chapter 8. Exponentiation and Orders Exercises for Chapter 8 Chapter 9. Euler's Ø-Function Exercises for Chapter 9 Chapter 10. Finding the Order of a mod c Exercises for Chapter 10 Chapter 11. Primality Testing Chapter 12. The RSA Cipher System Chapter 13. Primitive Roots mod p Chapter 14. Polynomials Chapter 15. Tables of Indices mod p Chapter 16. Brahmagupta's Formula and Hypernumbers Chapter 17. Modules of Hypernumbers Chapter 18. A Canonical Form for Modules of Hypernumbers Chapter 19. Solution of [omitted] Chapter 20. Proof of the Theorem of Chapter 19 Chapter 21. Euler's Remarkable Discovery Chapter 22. Stable Modules Chapter 23. Equivalence of Modules Chapter 24. Signatures of Equivalence Classes Chapter 25. The Main Theorem Chapter 26. Modules That Become Principal When Squared Chapter 27. The Possible Signatures for Certain Values of A Chapter 28. The Law of Quadratic Reciprocity Chapter 29. Proof of the Main Theorem Chapter 30. The Theory of Binary Quadratic Forms Chapter 31. Composition of Binary Quadratic Forms Appendix. Cycles of Stable Modules Answers to Exercises Bibliography Index Back Cover Higher Arithmetic Explains Number Theory In A Way That Gives Deductive Reasoning, Including Algorithms And Computations, The Central Role. Hands-on Experience With The Application Of Algorithms To Computational Examples Enables Students To Master The Fundamental Ideas Of Basic Number Theory. This Is A Worthwhile Goal For Any Student Of Mathematics And An Essential One For Students Interested In The Modern Applications Of Number Theory.--jacket. Numbers -- The Problem A + B = -- Congruences -- Double Congruences And The Euclidean Algorithm -- The Augmented Euclidean Algorithm -- Simultaneous Congruences -- The Fundamental Theorem Of Arithmetic -- Exponentiation And Orders -- Euler's 0-funtion -- Finding The Order Of A Mod C -- Primality Testing -- The Rsa Cipher System -- Primitive Roots Mod P -- Polynomials -- Tables Of Indices Mod P. Brahmagupta's Formule And Hypernumbers -- Modules Of Hypernumbers -- A Canonical Form For Modules Of Hypernumbers -- Solution Of A + B = -- Proof Of The Theorem Of Chapter 19 -- Euler's Remarkable Discovery -- Stable Modules -- Equivalence Of Modules -- Signatures Of Equivalence Classes -- The Main Theorem -- Modules That Become Principal When Squared -- The Possible Signatures For Certain Values Of A -- The Law Of Quadratic Reciprocity -- Proof Of The Main Theorem -- The Theory Of Binary Quadratic Forms -- Composition Of Binary Quadratic Forms. Harold M. Edwards. Includes Bibliographical References (p. 207) And Index. Following Gauss in his first systematic treatise on it in 1801, Edwards (emeritus, mathematics, New York U.) prefers higher arithmetic to number theory as the name for the general study of specific relations among whole numbers. Theory is about thinking, he explains, and arithmetic is about doing, and he found in his courses that students would rather do calculations than listen to him talk about them. Therefore, he includes many exercises in this introduction for readers who do not necessarily have a deep background in mathematics. Annotation 2008 Book News, Inc., Portland, OR (booknews.com)