A Friendly Introduction to Number Theory / Edition 3

A Friendly Introduction to Number Theory / Edition 3

by Joseph H. Silverman
     
 

ISBN-10: 0131861379

ISBN-13: 9780131861374

Pub. Date: 03/21/2005

Publisher: Prentice Hall

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

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.

Product Details

ISBN-13:
9780131861374
Publisher:
Prentice Hall
Publication date:
03/21/2005
Edition description:
REV
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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