Elementary number theory

Index......Page
Front Matter......Page 1
Publisher's Preface......Page 3
Preface......Page 4
Contents......Page 7
Part One - Foundations of Number Theory......Page 9
I. The Greatest Common Divisor of Two Numbers......Page 10
II. Prime Numbers and Factorization into Prime Factors......Page 17
III. The Greatest Common Divisor of Several Numbers......Page 25
IV. Number-Theoretic Functions......Page 28
V. Congruences......Page 36
VI. Quadratic Residues......Page 52
VII. Pell's Equation......Page 75
Part Two - Brun's Theorem and Dirichlet's Theorem......Page 84
Introduction......Page 85
I. Some Elementary Inequalities of Prime-Number Theory......Page 86
II. Brun's Theorem on Prime Pairs......Page 92
§1. Further Theorems on Congruences......Page 102
§2. Characters......Page 107
§3. L-Series......Page 113
§4. Dirichlet's Proof......Page 123
Part Three - Decomposition into Two, Three, and Four Squares......Page 124
Introduction......Page 125
I. Farey Fractions......Page 127
II. Decomposition into Two Squares......Page 131
Introduction......Page 137
§1. Lagrange's Theorem......Page 138
§2. Determination of the Number of Solutions......Page 142
§1. Equivalence of Quadratic Forms......Page 147
§2. A Necessary Condition for Decomposability into Three Squares......Page 157
§3. The Necessary Condition is Sufficient......Page 158
Part Four - Class Number of Binary Quadratic Forms......Page 161
Introduction......Page 162
I. Factorable and Unfactorable Forms......Page 165
II. Classes of Forms......Page 167
III. The Finiteness of the Class Number......Page 170
IV. Primary Representations by Forms......Page 175
V. Representations of h(d) in terms of K(d)......Page 186
VI. Gaussian Sums......Page 192
Introduction......Page 197
§1. Kronecker's Proof......Page 198
§2. Schur's Proof......Page 202
§3. Merten's Proof......Page 208
VII. Reduction to Fundamental Discriminants......Page 214
VIII. The Determination of K(d) for Fundamental Discriminants......Page 216
IX. Final Formulas for the Class Number......Page 222
Appendix - Exercises......Page 225
Exercises for Chapter I......Page 226
Exercises for Chapter II......Page 227
Exercises for Chapter IV......Page 228
Exercises for Chapter V......Page 229
Exercises for Chapter VI......Page 231
Exercises for Chapter VII......Page 233
Exercises for Part Two......Page 235
Exercises for Chapter II......Page 236
Exercises for Chapter III......Page 237
Exercises for Part Three......Page 240
Exercises for Chapter III......Page 241
Exercises for Chapter IV......Page 242
Index......Page 244
Index of Conventions......Page 245
Index of Symbols......Page 246