Introduction to Number Theory

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Title Contents Preface 1. Introduction 1.1 What Is Number Theory? 1.2 Prerequisites 1.3 How to Use this Book 2. Divisibility and Primes 2.1 Introduction 2.2 Divisibility 2.3 The Greatest Common Divisor 2.4 Unique Factorization Appendix A: Euler's Proof of the lnfinitude of Primes 3. Congruences 3.1 Introduction 3.2 Basic Properties of Congruences 3.3 Some Special Congruences 3.4 Solving Polynomial Congruences, I 3.5 Solving Polynomial Congruences, II 3.6 Primitive Roots 3.7 Congruences -- Some Historical Notes 4. The Law of Quadratic Reciprocity 4.1 Introduction 4.2 Basic Properties of Quadratic Residues 4.3 The Gauss Lemma 4.4 The Law of Quadratic Reciprocity 4.5 Applications to Diophantine Equations 5. Arithmetic Functions 5.1 Introduction 5.2 Multiplicative Arithmetic Functions 5.3 The Möbius Inversion Formula 5.4 Perfect and Amicable Numbers 6. A Few Diophantine Equafions 6.1 Introduction 6.2 The Equation x^2 + y^2 = z^2 6.3 The Equation x^4 + y^4 = z^2 6.4 The Equation x^2 + y^2 = n 6.5 The Equation x^2 + y^2 + z^2 + w^2 = n 6.6 Pell's Equation: x^2 - d y^2 = 1 Appendix B: Diophantine Approximations Introduction to Chapters 7-11 7. The Gaussian Integers 7.1 Introduction 7.2 The Fundamental Theorem of Arithmetic in the Gaussian Integers 7.3. The Two-Square Problem Revisited 8. Arithmetic in Quadratic Fields 8.1 Introduction 8.2 Quadratic Fields 8.3 The Integers of a Quadratic Field 8.4 Binary Quadratic Forms 8.5 Modules 8.6 The Coefficient Ring of a Module 8.7 The Unit Theorem 8.8 Computing Elements of a Given Norm in a Module 9. Factorization Theory in Quadratic Fields 9.1 The Failure of Unique Factorization 9.2 Generalized Congruences and the Norm of a Module 9.3 Products and Sums of Modules 9.4 The Fundamental Factorization Theorem 9.5 The Prime Modules Belonging to O 9.6 Finiteness of the Class Number 10. Applications of the Factorization Theory to Diophantine Equations 10.1 The Diophantine Equation y^2 = x^3 + k 10.2 Proof of Fermat's Last Theorem for n = 3 10.3 Norm Form Equations 11. The Representation of Integers by Binary Quadratic Forms 11.1 Equivalence of Forms 11.2 Strict Similarity of Modules 11.3 The Correspondence Between Modules and Forms 11.4 The Representation of Integers by Binary Quadratic Forms 11.5 Composition Theory for Binary Quadratic Forms Tables Notation Diophantine Equations Index