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