A Friendly Introduction to Number Theory / Edition 3

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Upper Saddle River, NJ 2005 Hard cover 3rd ed. New in new dust jacket. Sewn binding. Cloth over boards. 448 p. Contains: Illustrations. Audience: General/trade.

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Overview

Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers.

Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on “Continued Fractions, Square Roots and Pell’s Equation.” Contains additional historical material, including material on Pell’s equation and the Chinese Remainder Theorem.

A useful reference for mathematics teachers.

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Editorial Reviews

Booknews
Silverman (Brown U.) originally wrote the book as a text for a course designed to attract non-science majors with little interest in pursuing the standard calculus sequence, and convince them to study some college mathematics. He expects readers to have some facility with high school algebra and access to a calculator, though he points out that those who know how to program a computer have great fun generating reams of data and implementing assorted algorithms. He mentions concepts from calculus now and then, but does not lay them down as barriers to cross. The first edition appeared in 1997. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780131861374
  • Publisher: Prentice Hall
  • Publication date: 3/21/2005
  • Edition description: REV
  • Edition number: 3
  • Pages: 448
  • Product dimensions: 6.20 (w) x 9.20 (h) x 1.10 (d)

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

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