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.
Author(s): Harold M. Edwards
Series: Student Mathematical Library, 45
Publisher: American Mathematical Society
Year: 2008
Language: English
Pages: C, xii, 210, B
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
