| Chapter 1 Introduction 1.1 What is number theory? 1.2 Algebraic properties of the set of integers 1.3 Types of proofs, and some examples . 1.4 Representation systems for the integers 1.5 The early history of number theory . Chapter 2 Unique Factorization and the GCD 2.1 The greatest common divisor . . 2.2 Unique factorization in other domains 2.3 The linear Diophantine equation . 2.4 The least common multiple . . . Chapter 3 Congruences and the Ring Zm 3.1 Congruence and residue classes 3.2 Complete and reduced residue systems; Euler's ~-function 3.3 Linear congruences 3.4 Higher-degree polynomial congruences 3.5 Thep-adic fields Chapter 4 Primitive Roots and the Group Um 4.1 Primitive roots. 4.2 The structure of U,, 4.3 nth power residues 4.4 An application to Fermat's equation Chapter 5 Quadratic Residues 5.1 Introduction 5.2 Quadratic residues of primes, and the Legendre symbol . 5.3 The law of quadratic reciprocity 5.4 The Jacobi symbol 5.5 Factorization of large integers Chapter 6 Number-Theoretic Functions and the Distribution of Primes 6.1 Introduction 6.2 The M6bius function 6.3 The function Ix] 6.4 The symbos"O", "o", “<<”, and“～” 6.5 The sieve of Eratosthenes 6.6 Sums involving primes 6.7 The true order of 7(x) 6.8 Primes in arithmetic progressions 6.9 Bertrand's hypothesis 6.10 The order of magnitude of p, a, and 6.11 Average order of magnitude 6.12 Brun's theorem on twin primes Chapter 7 Sums of Squares 7.1 Preliminaries 7.2 Primitive representations as a sum of two squares 7.3 The total number of representations 7.4 Sums of three squares 7.5 Sums of four squares 7.6 Waring's problem Chapter 8 Quadratic Equations and Quadratic Fields 8.1 Legendre's theorem 8.2 Pell's equation 8.3 Algebraic number fields and algebraic integers 8.4 Arithmetic in quadratic fields Chapter 9 Diophantine Approximation 9.1 Farey sequences and Hurwitz's theorem 9.2 Best approximations to a real number 9.3 Infinite continued fractions 9.4 Quadratic irrationalities 9.5 Applications to PelFs equation and to factorization 9.6 Equivalence of numbers 9.7 The transcendence of e Bibliography Appendix Index |
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