Solved and Unsolved Problems in Number Theory

Contents

Chapter I: From Perfect Numbers to the Quadratic Reciprocity Law

1 Perfect numbers
2 Euclid
3 Euler's converse proved
4 Euclid's algorithm
5 Cataldi and others
6 The prime number theorem
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 reciprocity law
20 The prime divisors of $n^2 +a$

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 reciprocity law a deep theorem?
26 Congruential equations with a prime modulus
27 Euler's $\phi$ function
28 Primitive roots with a prime modulus
29 $\mathfrak{M}_{p}$ 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 $\mathfrak{M}_{m}$
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 $\sqrt 2$ 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
56 A generalization and binary quadratic forms
57 Some applications
58 The significance of Fermat's equation
59 The main theorem
60 An algorithm
61 Continued fractions for $\sqrt N$
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
75 The second progress report begins
76 On judging conjectures
77 On judging conjectures, II
78 Subjective judgement, the creation of conjectures and inventions
79 Fermat's last "Theorem," III
80 Computing and algorithms
81 $\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)$ and all that
82 1993

Appendix

Statement on fundamentals

Table of definitions

References

Index