A Friendly Introduction to Number Theory
Pearson (Verlag)
978-0-321-81619-1 (ISBN)
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Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and seven books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.
Preface
Flowchart of Chapter Dependencies
Introduction
1. What Is Number Theory?
2. Pythagorean Triples
3. Pythagorean Triples and the Unit Circle
4. Sums of Higher Powers and Fermat’s Last Theorem
5. Divisibility and the Greatest Common Divisor
6. Linear Equations and the Greatest Common Divisor
7. Factorization and the Fundamental Theorem of Arithmetic
8. Congruences
9. Congruences, Powers, and Fermat’s Little Theorem
10. Congruences, Powers, and Euler’s Formula
11. Euler’s Phi Function and the Chinese Remainder Theorem
12. Prime Numbers
13. Counting Primes
14. Mersenne Primes
15. Mersenne Primes and Perfect Numbers
16. Powers Modulo m and Successive Squaring
17. Computing kth Roots Modulo m
18. Powers, Roots, and “Unbreakable” Codes
19. Primality Testing and Carmichael Numbers
20. Squares Modulo p
21. Quadratic Reciprocity
22. Proof of Quadratic Reciprocity
23. Which Primes Are Sums of Two Squares?
24. Which Numbers Are Sums of Two Squares?
25. Euler’s Phi Function and Sums of Divisors
26. Powers Modulo p and Primitive Roots
27. Primitive Roots and Indices
28. The Equation X4 + Y4 = Z4
29. Square–Triangular Numbers Revisited
30. Pell’s Equation
31. Diophantine Approximation
32. Diophantine Approximation and Pell’s Equation
33. Number Theory and Imaginary Numbers
34. The Gaussian Integers and Unique Factorization
35. Irrational Numbers and Transcendental Numbers
36. Binomial Coefficients and Pascal’s Triangle
37. Fibonacci’s Rabbits and Linear Recurrence Sequences
38. Cubic Curves and Elliptic Curves
39. Elliptic Curves with Few Rational Points
40. Points on Elliptic Curves Modulo p
41. Torsion Collections Modulo p and Bad Primes
42. Defect Bounds and Modularity Patterns
43. Elliptic Curves and Fermat’s Last Theorem
Index
*44. The Topsy-Turvey World of Continued Fractions [online]
*45. Continued Fractions, Square Roots, and Pell’s Equation [online]
*46. Generating Functions [online]
*47. Sums of Powers [online]
*A. A List of Primes [online]
*These chapters are available online.
| Erscheint lt. Verlag | 15.3.2012 |
|---|---|
| Sprache | englisch |
| Maße | 204 x 234 mm |
| Gewicht | 710 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 0-321-81619-6 / 0321816196 |
| ISBN-13 | 978-0-321-81619-1 / 9780321816191 |
| Zustand | Neuware |
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