**Student Mathematical Library**

Volume: 45;
2008;
210 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-0-8218-4439-7
Product Code: STML/45**

List Price: $45.00

Individual Price: $36.00

**Electronic ISBN: 978-1-4704-2153-3
Product Code: STML/45.E**

List Price: $42.00

Individual Price: $33.60

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#### Supplemental Materials

# Higher Arithmetic: An Algorithmic Introduction to Number Theory

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*Harold M. Edwards*

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.

#### Reviews & Endorsements

Clean and elegant in
the way it communicates with the reader, the mathematical spirit of this book
remains very close to that of C.F. Gauss in his 1801 *Disquisitiones
Arithmeticae*, almost as though Gauss had revised that classic for
21st-century readers.

-- CHOICE Magazine

...takes the reader on a colorful journey...

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Higher Arithmetic: An Algorithmic Introduction to Number Theory

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Chapter 1. Numbers 114 free
- Chapter 2. The Problem [omitted] 720
- Chapter 3. Congruences 1124
- Chapter 4. Double Congruences and the Euclidean Algorithm 1730
- Chapter 5. The Augmented Euclidean Algorithm 2336
- Chapter 6. Simultaneous Congruences 2942
- Chapter 7. The Fundamental Theorem of Arithmetic 3346
- Chapter 8. Exponentiation and Orders 3750
- Chapter 9. Euler's Ø-Function 4356
- Chapter 10. Finding the Order of a mod c 4558
- Chapter 11. Primality Testing 5164
- Chapter 12. The RSA Cipher System 5770
- Chapter 13. Primitive Roots mod p 6174
- Chapter 14. Polynomials 6780
- Chapter 15. Tables of Indices mod p 7184
- Chapter 16. Brahmagupta's Formula and Hypernumbers 7790
- Chapter 17. Modules of Hypernumbers 8194
- Chapter 18. A Canonical Form for Modules of Hypernumbers 87100
- Chapter 19. Solution of [omitted] 93106
- Chapter 20. Proof of the Theorem of Chapter 19 99112
- Chapter 21. Euler's Remarkable Discovery 113126
- Chapter 22. Stable Modules 119132
- Chapter 23. Equivalence of Modules 123136
- Chapter 24. Signatures of Equivalence Classes 129142
- Chapter 25. The Main Theorem 135148
- Chapter 26. Modules That Become Principal When Squared 137150
- Chapter 27. The Possible Signatures for Certain Values of A 143156
- Chapter 28. The Law of Quadratic Reciprocity 149162
- Chapter 29. Proof of the Main Theorem 153166
- Chapter 30. The Theory of Binary Quadratic Forms 155168
- Chapter 31. Composition of Binary Quadratic Forms 163176
- Appendix. Cycles of Stable Modules 169182
- Answers to Exercises 179192
- Bibliography 207220
- Index 209222
- Back Cover Backcover1226