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Addison-Wesley / Prentice Hall

Mathematics

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Friendly Introduction to Number Theory, A, 3/E
Joseph H. Silverman

ISBN-10: 0131861379
ISBN-13: 9780131861374

Publisher: Prentice Hall
Copyright: 2006
Format: Cloth; 448 pp
Published: 03/21/2005
Status: Instock

Suggested retail price: $112.00
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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.

• A low-key introduction to Number Theory – Enables students to explore an area of math different from standard calculus sequences.

• Five basic steps emphasized – Experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.

– Encourages students to make mathematical discovers on their own through use of open-ended problems. Ex.___

• RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured, enabling students to see real-life applications of mathematics.

• Proof of Fermat's Last theorem by Andrew Wiles – Provides overview, introducing students to one of the most significant mathematical achievements of the 20th century.

• A new chapter that introduces the theory of continued fractions – Includes the recursion formula for convergents and the difference of successive convergents (Ch. 39, “The Topsy-Turvy World of Continued Fractions”)

• 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 (Ch. 38, “Oh, What a Beautiful Function”).

• A new chapter on “Continued Fractions, Square Roots and Pell’s Equation” (Ch. 40) – A continuation of the previous chapter, inlcuding a discussion of periodicity of continued fractions for quadratic irrationalities and the relationship between such continued fraction and solutions to Pell’s equation.

• Additional historical material – Includes material on Pell’s equation and the Chinese Remainder Theorem.

• New exercises added to existing chapters.

• Some proofs have been rewritten for added clarity.

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 k^th 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