A Friendly Introduction to Number Theory, 3/e (IE-Paperback)
暫譯: 友善的數論入門 (第三版)

Joseph H. Silverman

A Friendly Introduction to Number Theory, 3/e (IE-Paperback)
  • 出版商: Prentice Hall
  • 出版日期: 2005-03-31
  • 售價: $700
  • 貴賓價: 9.8 折 $686
  • 語言: 英文
  • 頁數: 448
  • 裝訂: Hardcover
  • ISBN: 0131861379
  • ISBN-13: 9780131861374

    • 已過版

買這商品的人也買了...

相關主題

商品描述

Description

For courses in Elementary Number Theory for math majors, for mathematics education students, and for Computer Science students.

 

This introductory undergraduate text is designed to entice a wide variety of majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal 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.

 

Table of Contents

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 Numbers8

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. Euler’s Phi Function and Sums of Divisors

21. Powers Modulo p and Primitive Roots

22. Primitive Roots and Indices

23. Squares Modulo p

24. Is —1 a Square Modulo p? Is 2?

25. Quadratic Reciprocity

26. Which Primes Are Sums of Two Squares?

27. Which Numbers Are Sums of Two Squares?

28. The Equation X4 + Y 4 = 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. Oh, What a Beautiful Function

39. The Topsy-Turvy World of Continued Fractions

40. Continued Fractions, Square Roots and Pell’s Equation

41. Generating Functions

42. Sums of Powers

43. Cubic Curves and Elliptic Curves

44. Elliptic Curves with Few Rational Points

45. Points on Elliptic Curves Modulo p

46. Torsion Collections Modulo p and Bad Primes

47. Defect Bounds and Modularity Patterns

48. Elliptic Curves and Fermat’s Last Theorem

Further Reading

A. Factorization of Small Composite Integers

B. A List of Primes

商品描述(中文翻譯)

描述

這本針對數學主修、數學教育學生以及計算機科學學生的初級數論課程的入門本科教材，旨在吸引各種專業的學生學習數學，同時教導他們數學思考。從基本的高中代數開始，讀者逐步從基本代數進入到積極進行數學研究的階段，並窺探當前數學的前沿。寫作風格非正式，包含許多數值範例，這些範例被分析以尋找模式並用於提出猜想。重點在於證明定理所使用的方法，而非具體的結果。

目錄

1. 什麼是數論？
2. 畢氏三元組
3. 畢氏三元組與單位圓
4. 高次方的和與費馬最後定理
5. 可除性與最大公因數
6. 線性方程與最大公因數
7. 因式分解與算術基本定理
8. 同餘
9. 同餘、冪與費馬小定理
10. 同餘、冪與歐拉公式
11. 歐拉的φ函數與中國剩餘定理
12. 質數
13. 計數質數
14. 梅森質數
15. 梅森質數與完美數
16. 模m的冪與連續平方
17. 計算模m的k次根
18. 冪、根與「無法破解」的代碼
19. 質性測試與卡邁克爾數
20. 歐拉的φ函數與因數和
21. 模p的冪與原根
22. 原根與指數
23. 模p的平方
24. -1是模p的平方嗎？2呢？
25. 二次互反律
26. 哪些質數是兩個平方的和？
27. 哪些數是兩個平方的和？
28. 方程X^4 + Y^4 = Z^4
29. 平方三角數的再探討
30. 佩爾方程
31. 迪奧方程的近似
32. 迪奧方程的近似與佩爾方程
33. 數論與虛數
34. 高斯整數與唯一分解
35. 無理數與超越數
36. 二項係數與帕斯卡三角形
37. 費波那契的兔子與線性遞歸序列
38. 哦，何等美妙的函數
39. 連分數的顛倒世界
40. 連分數、平方根與佩爾方程
41. 生成函數
42. 冪的和
43. 三次曲線與橢圓曲線
44. 具有少量有理點的橢圓曲線
45. 模p的橢圓曲線上的點
46. 模p的扭曲集合與壞質數
47. 缺陷界限與模性模式
48. 橢圓曲線與費馬最後定理

進一步閱讀

A. 小合成整數的因式分解
B. 質數列表

類似商品

最後瀏覽商品 (4)