A friendly introduction to number theory
著者
書誌事項
A friendly introduction to number theory
Pearson Education, c2013
4th ed
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注記
Includes index
内容説明・目次
内容説明
A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce readers 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, readers 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 analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
目次
Preface
Flowchart of Chapter Dependencies
Introduction
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. Quadratic Reciprocity
22. Proof of Quadratic Reciprocity
23. Which Primes Are Sums of Two Squares?
24. Which Numbers Are Sums of Two Squares?
25. Euler's Phi Function and Sums of Divisors
26. Powers Modulo p and Primitive Roots
27. Primitive Roots and Indices
28. The Equation X4 + Y4 = 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. Cubic Curves and Elliptic Curves
39. Elliptic Curves with Few Rational Points
40. Points on Elliptic Curves Modulo p
41. Torsion Collections Modulo p and Bad Primes
42. Defect Bounds and Modularity Patterns
43. Elliptic Curves and Fermat's Last Theorem
Index
*44. The Topsy-Turvey World of Continued Fractions [online]
*45. Continued Fractions, Square Roots, and Pell's Equation [online]
*46. Generating Functions [online]
*47. Sums of Powers [online]
*A. A List of Primes [online]
*These chapters are available online.
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