A Friendly Introduction to Number Theory, 4/e (Paperback)
暫譯: 友善的數論入門（第4版，平裝本）

Joseph H. Silverman

A Friendly Introduction to Number Theory, 4/e (Paperback)

相關主題

商品描述

本書序言

●Many new exercises appear throughout the text.
●Content updates throughout include:。A new chapter on mathematical induction (Chapter 26).
。Some material on proof by contradiction has been moved forward to Chapter 8.
。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).
。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.

本書特色

For one-semester undergraduate courses in Elementary Number Theory 

A Friendly Introduction to Number Theory, 4th 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 analysed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.

●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 emphasised 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.

商品描述(中文翻譯)

本書序言

●本書中出現了許多新的練習題。

●內容更新包括：新增一章有關數學歸納法的內容（第26章）。

。部分關於反證法的材料已提前移至第8章。

。原始根的章節（第28至29章）已移至二次互反和平方和的章節之後（第20至25章）。

。第22章現在包含了Jacobi符號的二次互反部分的證明，其餘部分作為練習題。

。二次互反現在已完整證明。(-1/p)和(2/p)的證明仍然保留在第21章，並且有一個新章節（第23章）提供了Eisenstein對於(p/q)(q/p)的證明。第23章的難度顯著高於之前的章節，可以在不影響後續章節的情況下省略。

。作為原始根的應用，第28章討論了Costas陣列的構造。

。第39章包括了一個證明，證明當p對5取模時同餘於1或4時，Fibonacci序列的周期模p會整除p - 1。

本書特色

適用於一學期的本科生初等數論課程

《數論入門（第4版）》旨在通過對數論這一特定方面的詳細研究，向學生介紹數學的整體主題和方法論。從基本的高中代數開始，學生逐漸被引導到主動進行數學研究的階段，同時窺見當前數學的前沿。這本書的寫作風格適合本科生，並包含許多數值例子，這些例子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，而不是具體的結果。

●50個短章節為教師和學生提供了靈活性和選擇。此版本中包含了章節依賴關係的流程圖。

●全書強調五個基本步驟，以幫助讀者發展穩健的思考過程：。實驗

。模式識別

。假設形成

。假設測試

。正式證明

●本書介紹了RSA加密系統、橢圓曲線和費馬最後定理，展示了數學的現實應用。

作者簡介

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.

作者簡介(中文翻譯)

約瑟夫·H·西爾弗曼是布朗大學的數學教授。他在布朗大學獲得學士學位（Sc.B.），並在哈佛大學獲得博士學位（Ph.D.），之後曾在麻省理工學院和波士頓大學任職，並於1988年加入布朗大學的教職。他在數論、橢圓曲線、算術幾何、算術動力系統和密碼學等領域發表了超過100篇經過同行評審的研究文章和七本書籍。他是一位備受推崇的教師，曾獲得布朗大學和美國數學協會的教學獎項，以及美國數學學會頒發的斯蒂爾數學表述獎。他指導了超過25位博士生的論文，是NTRU Cryptosystems, Inc.的共同創辦人，並曾擔任美國數學學會理事會和執行委員會的當選成員。

目錄大綱

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. Is -1 a Square Modulo p? Is 2?
22. Quadratic Reciprocity
23. Proof of Quadratic Reciprocity
24. Which Primes Are Sums of Two Squares?
25. Which Numbers Are Sums of Two Squares?
26. As Easy as One, Two, Three
27. Euler's Phi Function and Sums of Divisors
28. Powers Modulo p and Primitive Roots
29. Primitive Roots and Indices
30. The Equation X4 + Y4 = Z4
31. Square–Triangular Numbers Revisited
32. Pell's Equation
33. Diophantine Approximation
34. Diophantine Approximation and Pell's Equation
35. Number Theory and Imaginary Numbers
36. The Gaussian Integers and Unique Factorization
37. Irrational Numbers and Transcendental Numbers
38. Binomial Coefficients and Pascal's Triangle
39. Fibonacci's Rabbits and Linear Recurrence Sequences
40. Oh, What a Beautiful Function
41. Cubic Curves and Elliptic Curves
42. Elliptic Curves with Few Rational Points
43. Points on Elliptic Curves Modulo p
44. Torsion Collections Modulo p and Bad Primes
45. Defect Bounds and Modularity Patterns
46. Elliptic Curves and Fermat's Last Theorem
47 The Topsy-Turvy World of Continued Fractions [online]
48 Continued Fractions and Pell's Equation [online]
49 Generating Functions [online]
50 Sums of Powers [online]
A Factorization of Small Composite Integers [online]
B A List of Primes [online]

目錄大綱(中文翻譯)

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. Is -1 a Square Modulo p? Is 2?

22. Quadratic Reciprocity

23. Proof of Quadratic Reciprocity

24. Which Primes Are Sums of Two Squares?

25. Which Numbers Are Sums of Two Squares?

26. As Easy as One, Two, Three

27. Euler's Phi Function and Sums of Divisors

28. Powers Modulo p and Primitive Roots

29. Primitive Roots and Indices

30. The Equation X4 + Y4 = Z4

31. Square–Triangular Numbers Revisited

32. Pell's Equation

33. Diophantine Approximation

34. Diophantine Approximation and Pell's Equation

35. Number Theory and Imaginary Numbers

36. The Gaussian Integers and Unique Factorization

37. Irrational Numbers and Transcendental Numbers

38. Binomial Coefficients and Pascal's Triangle

39. Fibonacci's Rabbits and Linear Recurrence Sequences

40. Oh, What a Beautiful Function

41. Cubic Curves and Elliptic Curves

42. Elliptic Curves with Few Rational Points

43. Points on Elliptic Curves Modulo p

44. Torsion Collections Modulo p and Bad Primes

45. Defect Bounds and Modularity Patterns

46. Elliptic Curves and Fermat's Last Theorem

47 The Topsy-Turvy World of Continued Fractions [online]

48 Continued Fractions and Pell's Equation [online]

49 Generating Functions [online]

50 Sums of Powers [online]

A Factorization of Small Composite Integers [online]

B A List of Primes [online]

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