An Introduction to the Theory of Numbers

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