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Higher Arithmetic: An Algorithmic Introduction to Number Theory

Harold M. Edwards New York University, New York, NY
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Softcover 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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List Price: $67.50 Click above image for expanded view Higher Arithmetic: An Algorithmic Introduction to Number Theory Harold M. Edwards New York University, New York, NY Available Formats:  Softcover 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 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$67.50
• Book Details

Student Mathematical Library
Volume: 452008; 210 pp
MSC: Primary 11;

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.

• Chapters
• Chapter 1. Numbers
• Chapter 2. The problem $A\square + B = \square$
• Chapter 3. Congruences
• Chapter 4. Double congruences and the Euclidean algorithm
• Chapter 5. The augmented Euclidean algorithm
• Chapter 6. Simultaneous congruences
• Chapter 7. The fundamental theorem of arithmetic
• Chapter 8. Exponentiation and orders
• Chapter 9. Euler’s $\phi$-function
• Chapter 10. Finding the order of $a\bmod c$
• Chapter 11. Primality testing
• Chapter 12. The RSA cipher system
• Chapter 13. Primitive roots $\bmod \, p$
• Chapter 14. Polynomials
• Chapter 15. Tables of indices $\bmod \, p$
• 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 $A\square + B = \square$
• 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

• Reviews

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 452008; 210 pp
MSC: Primary 11;

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.

• Chapters
• Chapter 1. Numbers
• Chapter 2. The problem $A\square + B = \square$
• Chapter 3. Congruences
• Chapter 4. Double congruences and the Euclidean algorithm
• Chapter 5. The augmented Euclidean algorithm
• Chapter 6. Simultaneous congruences
• Chapter 7. The fundamental theorem of arithmetic
• Chapter 8. Exponentiation and orders
• Chapter 9. Euler’s $\phi$-function
• Chapter 10. Finding the order of $a\bmod c$
• Chapter 11. Primality testing
• Chapter 12. The RSA cipher system
• Chapter 13. Primitive roots $\bmod \, p$
• Chapter 14. Polynomials
• Chapter 15. Tables of indices $\bmod \, p$
• 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 $A\square + B = \square$
• 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