Elementary Introduction to Number Theory

✍ Scribed by Calvin T. Long

• Publisher: Prentice-Hall
• Year: 1987
• Tongue: English
• Leaves: 304
• Edition: 3
• Category: Library

✦ Table of Contents

Title
Contents
Preface To The Third Edition
Preface To The Second Edition
Preface To The First Edition
1. Preliminary Considerations
1.1. Summation and multiplication notation
1.2. Inductive reasoning and the Fibonacci sequence
1.3. The postulates of mathematical induction and well-ordering
1.4. Mathematical induction
1.5. The well-ordering principle
1.6. Equivalence of the principles of induction and well-ordering
1.7. The division algorithm
1.8. Positional notation
1.9. Computational complexity
2. Divisibility Properties of Integers
2.1. Basic properties
2.2. The greatest common divisor
2.3. The Euclidean algorithm
2.4. The least common multiple
2.5. The fundamental theorem of arithmetic
2.6. Pythagorean triples
2.7. The greatest integer function
3. Prime Numbers
3.1. The sieve of Eratosthenes
3.2. The infinitude of primes
3.3. The prime number theorem
3.4. Mersenne, Fermat, and perfect numbers
4. Congruences
4.1. Basic definitions and properties
4.2. Special divisibility criteria
4.3. Reduced residue systems and the Euler phi-function
4.4. Pseudoprimes and tests for primality
4.5. Some contacts with abstract algebra
5. Conditional Congruences
5.1. Linear congruences
5.2. The Chinese remainder theorem
5.3. Polynomial congruences of degree greater than 1
5.4. Theorems of Lagrange and Wilson
5.5. Quadratic congruences
5.6. The quadratic reciprocity law of Gauss
5.7. Primitive roots, indices, and power residues
6. Cryptography
6.1. Caesar ciphers
6.2. Exponentiation ciphers
6.3. Public key encryption systems
7. Sums of Squares
7.1. Sums of two squares
7.2. Sums of more than two squares
7.3. Waring's theorem
8. Multiplicative Number-Theoretic Functions
8.1. Definitions
8.2. Notation
8.3. Multiplicative number-theoretic functions
8.4. The Möbius inversion formula
8.5. The Euler phi-functioon
8.6. Other inversion formulas
8.7. A further inversion formula
9. Simple Continued Fractions
9.1. Finite simple continued fractions
9.2. Convergents
9.3. Infinite simple continued fractions
9.4. Periodic simple continued fractions
9.5. Approximation of irrationals by rationals
References
Tables
Answers to Selected Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Index