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 classicDisquisitiones 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.
• Preface 10 • Chapter 1. Numbers 14 • Chapter 2. The Problem [omitted] 20 • Chapter 3. Congruences 24 • Chapter 4. Double Congruences and the Euclidean Algorithm 30 • Chapter 5. The Augmented Euclidean Algorithm 36 • Chapter 6. Simultaneous Congruences 42 • Chapter 7. The Fundamental Theorem of Arithmetic 46 • Chapter 8. Exponentiation and Orders 50 • Chapter 9. Euler's Ø - Function 56 • Chapter 10. Finding the Order of a mod c 58 • Chapter 11. Primality Testing 64 • Chapter 12. The RSA Cipher System 70 • Chapter 13. Primitive Roots mod p 74 • Chapter 14. Polynomials 80 • Chapter 15. Tables of Indices mod p 84 • Chapter 16. Brahmagupta's Formula and Hypernumbers 90 • Chapter 17. Modules of Hypernumbers 94 • Chapter 18. A Canonical Form for Modules of Hypernumbers 100 • Chapter 19. Solution of [omitted] 106 • Chapter 20. Proof of the Theorem of Chapter 19 112 • Chapter 21. Euler's Remarkable Discovery 126 • Chapter 22. Stable Modules 132 • Chapter 23. Equivalence of Modules 136 • Chapter 24. Signatures of Equivalence Classes 142 • Chapter 25. The Main Theorem 148 • Chapter 26. Modules That Become Principal When Squared 150 • Chapter 27. The Possible Signatures for Certain Values of A 156 • Chapter 28. The Law of Quadratic Reciprocity 162 • Chapter 29. Proof of the Main Theorem 166 • Chapter 30. The Theory of Binary Quadratic Forms 168 • Chapter 31. Composition of Binary Quadratic Forms 176 • Appendix. Cycles of Stable Modules 182 • Answers to Exercises 192 • Bibliography 220 • Index 222