An Introduction To The Theory Of Numbers (merrill Mathematics Series)

Title Preface Contents 1. Introduction 1-1 Nature of the Subject 1-2 Some Questions Considered 1-3 Problems 2. Divisibility 2-1 Introduction 2-2 Sundry Definitions 2-3 Elementary Theorems 2-4 Some Fundamental Principles 2-5 Basic Theorem 2-6 Mathematical Induction 2-7 Problems 2-8 Scales of Notation 2-9 Problems 2-10 Common Divisors 2-11 Euclid’s Algorithm 2-12 Linear Diophantine Equations 2-13 Problems 2-14 Greatest Common Divisor and Least Common Multiple 2-15 Number of Primes Infinite 2-16 Sieve of Eratosthenes 2-17 Unique Factorization 2-18 Problems 3. Congruences 3-1 Residue Classes 3-2 Congruence Symbol 3-3 Properties of Congruences 3-4 Problems 3-5 Euler's phi-Function 3-6 Fermat’s Theorem and Euler's Generalization 3-7 Pseudoprimes 3-8 Problems 3-9 Linear Congruences and Their Solution 3-10 Simple Continued Fractions 3-11 Wilson's Theorem 3-12 The Chinese Remainder Theorem 3-13 Problems 3-14 Identical and Conditional Congruences 3-15 Equivalent Congruences 3-16 Division of Polynomials, modulo m 3-17 Problems 3-18 Number of Solutions of a Congruence 3-19 Number of Solutions of Special Congruences 3-20 Number of Solutions of a Binomial Quadratic Congruence 3-21 Problems 3-22 Solution of the Congruence f(x) equiv 0 (mod m) 3-23 Polynomials Representing Primes 3-24 Problems 4. Some Significant Functions in the Theory of Numbers 4-1 The Greatest Integer Function 4-2 Problems 4-3 Generalization of Euler’s phi-Function 4-4 Functions tau(n) and sigma(n) 4-5 Problems 4-6 Perfect Numbers 4-7 Möbius mu-Function 4-8 Liouville's Function lambda(n) 4-9 Problems 4-10 Recurrence Formulae 4-11 Fibonacci’s and Lucas’ Sequences 4-12 Problems 5. Primitive Roots and lndices 5-1 Belonging to an Exponent 5-2 Problems 5-3 Primitive Roots 5-4 Obtaining Primitive Roots 5-5 Sum of Numbers Belonging to an Exponent 5-6 Further Consideration of Primitive Roots of p^n 5-7 Problems 5-8 Indices 5-9 Problems 6. Quadratic Congruences 6-1 A Quadratic Congruence 6-2 Quadratic Residue and Quadratic Nonresidue 6-3 Problems 6-4 Euler's Criterion 6-5 Legendre’s Symbol 6-6 The Quadratic Reciprocity Law 6-7 Problems 6-8 Another Proof of the Quadratic Reciprocity Law 6-9 The Jacobi Symbol 6-10 Generalized Quadratic Reciprocity Law 6-11 Problems 7. Elementary Considerations on the Distribution of Primes and Composites 7-1 Introduction 7-2 The O-notation 7-3 Problems 7-4 Bertrand's Postulate 7-5 Problems 7-6 Bounds for pi(x) 7-7 Remarks on the Prime Number Theorem 7-8 Primes in Arithmetical Progressions 7-9 Highly Composite Numbers 7-10 Relatively Highly Composite Numbers 7-11 Problems 8. Continued Fractions 8-1 Introduction 8-2 Finite Continued Fractions 8-3 Convergents and Their Limits 8-4 Problems 8-5 Representation of Irrational Numbers 8-6 Approximation by Rational Numbers 8-7 Problems 8-8 Quadratic Irrational Numbers 8-9 Periodic Continued Fractions 8-10 Problems 8-11 Pell's Equation 8-12 Problems 8-13 Farey Sequences 8-14 Problems 9. Certain Diophantine Equations and Sums of Squares 9-1 Introductory Remarks 9-2 The Pythagorean Equation 9-3 The Diophantine Equation x^2 + 2y^2 = z^2 9-4 Problems 9-5 Some Fourth Degree Diophantine Equations 9-6 Problems 9-7 Solution of the Equations X^4 - 2Y^4 = plusminus Z^2 9-8 Sum of Two Squares 9-9 Sum of Three Squares 9-10 Problems 9-11 Sum of Four Squares 9-12 Remarks on Waring's Problem 9-13 Problems Notes Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Bibliography Appendix Index