[Cover text]
THIS INTRODUCTION TO THE THEORY OF NUMBERS PLACES SPECIAL EMPHASIS ON BASIC CONCEPTS AND VARIETY OF TREATMENT THROUGHOUT. Its variety of approaches and methods is designed to show the richness of the theory of numbers.
The author emphasizes that theory can not be fully understood until it can be effectively applied. Accordingly, the book provides more than the usual number at exercises and problems.
Detailed exposition in the early chapters, especially in the first three, also aids understanding. The book includes many illustrative examples worked out in detail to reinforce the theory presented. It frequently gives several alternative methods of proof.
A series of Notes on each chapter covers (1) reference to additional material, (2) detailed proofs by mathematical induction, and (3) discussion of interesting sidelights.
RALPH G. AHCHIBALD (Ph. D, University of Chicago) is Professor of Mathematics at Queens College of the City University of New York. He has published a number of papers in mathematical journals in the United States and Canada.
Author(s): Ralph G. Archibald
Series: Merrill Mathematics Series
Publisher: Charles E. Merrlll Publishing Co.
Year: 1970
Language: English
Commentary: New scan of https://libgen.rs/book/index.php?md5=4B831870DE1DE4EB4A696D8408E52C7E
Pages: 305+xi
City: Columbus, Ohio
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
