A Friendly Introduction to Number Theory 4/E
- ISBN13:9781292027098
- 替代書名:數論學
- 出版社:PRENTICE-HALL;INC.
- 作者:SILVERMAN
- 裝訂/頁數:平裝/480頁
- 出版日:2014/01/01
商品簡介
For one-semester undergraduate courses in Elementary Number Theory.
A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet—number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
Features
50 short chapters provide flexibility and options for instructors and students. A flowchart of chapter dependencies is included in this edition.
Five basic steps are emphasized throughout the text to help readers develop a robust thought process:
Experimentation
Pattern recognition
Hypothesis formation
Hypothesis testing
Formal proof
RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured, showing the real-life applications of mathematics.
New to this Edition
There are a number of major changes in the Fourth Edition.
Many new exercises appear throughout the text.
A flowchart giving chapter dependencies is included to help instructors choose the most appropriate mix of topics for their students.
Content Updates
There is a new chapter on mathematical induction (Chapter 26).
Some material on proof by contradiction has been moved forward to Chapter 8. It is used in the proof that a polynomial of degree d has at most d roots modulo p. This fact is then used in place of primitive roots as a tool to prove Euler’s quadratic residue formula in Chapter 21. (In earlier editions, primitive roots were used for this proof.)
The chapters on primitive roots (Chapters 28–29) have been moved to follow the chapters on quadratic reciprocity and sums of squares (Chapters 20–25). The rationale for this change is the author’s experience that students find the Primitive Root Theorem to be among the most difficult in the book. The new order allows the instructor to cover quadratic reciprocity first, and to omit primitive roots entirely if desired.
Chapter 22 now includes a proof of part of quadratic reciprocity for Jacobi symbols, with the remaining parts included as exercises.
Quadratic reciprocity is now proved in full. The proofs for (-1/p) and (2/p) remain as before in Chapter 21, and there is a new chapter (Chapter 23) that gives Eisenstein’s proof for (p/q)(q/p). Chapter 23 is significantly more difficult than the chapters that precede it, and it may be omitted without affecting the subsequent chapters.
As an application of primitive roots, Chapter 28 discusses the construction of Costas arrays.
Chapter 39 includes a proof that the period of the Fibonacci sequence modulo p divides p - 1 when p is congruent to 1 or 4 modulo 5.
Number theory is a vast and sprawling subject, and over the years this book has acquired many new chapters. In order to keep the length of this edition to a reasonable size, Chapters 47, 48, 49, and 50 have been removed from the printed version of the book.
目次
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
44. The Topsy-Turvy World of Continud Fractions
45. Continued Fractions and Pell’s Equation
46. Generating Functions
47. Sums of Powers
48. Appendix: A List of Primes
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