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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.
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
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
A. Factorization of Small Composite Integers
B. A List of Primes