Table of Contents
Preface v
I The Division Algorithm 1
1 The Origin of Natural Numbers 1
Perfect Numbers and Amicable Numbers 7
2 The Mystery of Natural Numbers 10
Mosaic Geometry and Euler's Characteristic 16
3 The Division Algorithm 18
Mersenne Primes and Fermat Primes 24
4 The Greatest Common Divisor 32
Graham's Conjecture 37
5 The Fundamental Theorem of Arithmetic 39
Hubert's 8th Problem 47
Exercises 1 54
II The Concept of Congruence 55
6 The Concept of Congruence 55
Gauss' Disquisitiones Arithmeticae 59
7 Residue Classes and Residue Systems 62
Function [x] and the 3x + 1 Problem 66
8 The Fermat-Euler Theorem 70
The Euler Number and the Euler Prime 78
9 Fractions Expressed as Repeating Decimals 80
Möbius' Function 83
10 An Application to Cryptology 88
The Generalized Euler Function 92
Exercises 2 96
III Congruences 97
11 Qin Jiushao's Theorem 97
Fibonacci's Rabbits 103
12 Wilson's Theorem 106
A Theorem that Gauss Did Not Prove 112
13 The Diophantine Equation 115
The Pythagorean Triple 120
14 Lucas' Congruence 122
Covering Systems 129
15 The Truth of Primes 131
The Chain of Primes or Composite Numbers 137
Exercises 3 141
IV Quadratic Residues 143
16 Quadratic Congruences 143
Integers in the Gaussian Ring 147
17 The Legendre Symbol 150
Representing Integers as the Sum of Squares 155
18 The Law of Quadratic Reciprocity 160
N-Gonal Numbers and Fermat 163
19 The Jacobi Symbol 165
The Hadarnard Matrix and Hadamard's Conjecture 170
20 Congruences Modulo a Composite 172
Constructibility of the Regular 17-Gon 176
Exercises 4 179
V Nth Power Residues 181
21 Definition of Orders 181
Egyptian Fractions 184
22 The Existence of Primitive Roots 186
Artin's Conjecture 189
23 The Nth Power Residue 191
Pell's Equation 200
24 The Case of Composite Modulus 204
Diophantine Arrays 206
25 The Dirichlet Character 209
Three Special Exponent Sums 214
Exercises 5 219
VI Congruences Modulo Integer Powers 221
26 Bernoulli Numbers and Bernoulli Polynomials 221
The Kummer Congruence 226
27 The Wolstenholme Theorem 229
Elliptic Curves 233
28 Lehmer's Congruence 239
The abc Conjecture 248
29 Morley's Theorem and Jacobstahl's Theorem 254
Automorphic Forms and Modular Forms 263
30 Congruences on Harmonic Sums 267
Multinomial Coefficients of Non-Powers 271
VII Additive and Multiplicative Number Theory 277
31 New Waring's Problem 277
32 New Format's Last Theorem 286
33 Eider's Conjecture 295
34 The F-Perfect Number Problem 303
35 A New Congruent Number Problem 313
36 The ABCD Equation 322
Appendix A The List of Prime Numbers Less Than 10000 335
Bibliography 339
