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Product Code:  STML/45 
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Individual Price:  $36.00 
Electronic ISBN:  9781470421533 
Product Code:  STML/45.E 
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Book DetailsStudent Mathematical LibraryVolume: 45; 2008; 210 ppMSC: 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. Handson 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.ReadershipUndergraduates, graduate students, and research mathematicians interested in number theory.

Table of Contents

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

Answers to exercises


Additional Material

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 21stcentury readers.
CHOICE Magazine 
...takes the reader on a colorful journey...
Mathematical Reviews


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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. Handson 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.
Undergraduates, graduate students, and research mathematicians interested in number theory.

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

Answers to exercises

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 21stcentury readers.
CHOICE Magazine 
...takes the reader on a colorful journey...
Mathematical Reviews