Solved and Unsolved Problems in Number Theory

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PREFACE
Chapter I FROM PERFECT NUMBERS TO THE QUADRATIC RECIPROCITY LAW
1. Perfect Numbers
2. Euclid
3. Euler's Converse Proved
4. Euc1id's Algorithm
5. Cataldi and Others
6. The Prime Kumber Theorrm
7. Two Useful Theorems
8. Fermat and Others
9. Euler's Generalization Proved
10. Perfect Numbers, II
11. Euler and M_{31}
12. Many Conjectures and their Interrelations
13. Splitting the Primes into Equinumerous Classes
14. Euler's Criterion Formulated
15, Euler's Criterion Proved
16. Wilson's Theorem
17. Gauss's Criterion
18. The Original Legendre Symbol
19. The Reciproeity Law
20. The Prime Divisors of n²+a
Chapter II THE UNDERLYING STRUCTURE
21. The Residue Classes as an Invention
22. The Residue Classes as a Tool
23. The Residue Classes as a Group
24. Quadratic Residues
25. Is the Quadratic Reriproeity Law a Deep Theorem?
26. Congruential Equations with a Prime Modulus
27. Euler's φ Function
28. Primitive Roots with a Prime Modulus
29. M_p as a Cyclic Group
30. The Circular Parity Switch
31. Primitive Roots and Fermat Numbers
32. Artin's Conjectures
33. Questions Concerning Cycle Graphs
34. Answers Concerning Cycle Graphs
35. Factor Generators of M_m
36. Primes in Some Arithmetic Progressions and a General Divisibility Theorem
37. Scalar and Vector Indices
38. The Other Residue Classes
39. The Converse of Fermat's Theorem
40. Sufficient Conditions for Primality
Chapter III PYTHAGOREANISM AND ITS MANY CONSEQUENCES
41. The Pythagoreans
42. The Pythagorean Theorem
43. The √2 and the Crisis
44. The Effect upon Geometry
45. The Case for Pythagoreanism
46. Three Greek Problems
47. Three Theorems of Fermat
48. Fermat's Last "Theorem"
49. The Easy Case and Infinite Descent
50. Gaussian Integers and Two Applications
51. Algebraic Integers and Kummer's Theorem
52. The Restricted Case, Sophie Germain, and Wieferich
53. Euler's "Conjecture"
54. Sum of Two Squares
55. A Generalization and Geometric Number Theory
5G. A Generalization and Binary Quadratic Forms
57. Some Applications
58 The Significance of Fermat's Equatlon
59. The Main Theorem
60. An Algorithm
61. Continued Fractions for √N
62. From Archimedes to Lucas
63. The Lucas Criterion
64. A Probability Argument
65. Fibonacci Numbers and the Original Lucas Test
Appendix to Chapters I-III SUPPLEMENTARY COMMENTS, THEOREMS, AND EXERCISES
Chapter IV PROGRESS
66. Chapter I Fifteen Years Later
67. Artin's Conjectures, II
68. Cycle Graphs and Related Topics
69. Pseudoprimes and Primality
70. Fermat's Last "Theorem," II
71. Binary Quadratic Forms with Negative Discriminants
72. Binary Quadratic Forms with Positive Discriminants
73. Lucas and Pythagoras
74. The Progress Report Concluded
STATEMENT ON FUNDAMENTALS
TABLE OF DEFINITIONS
REFERENCES
INDEX