Number theory: an introduction

✍ Scribed by Don Redmond

• Publisher: M. Dekker
• Year: 1996
• Tongue: English
• Leaves: 776
• Series: Monographs and textbooks in pure and applied mathematics 201
• Edition: 1
• Category: Library

✦ Synopsis

This reference text provides a detailed introduction to number theory--demonstrating how other areas of mathematics enter into the study of the properties of natural numbers.

✦ Table of Contents

Title
Preface
Contents
A Historical Introduction
1. Primes and Divisibility
1.1. Introduction
1.2. Divisibility
1.3. Prime numbers
1.4. Unique factorization
1.5. Some factorization methods, I
Additional problems
2. Congruences
2.1. Congruences
2.2. The Euler phi function
2.3. Congruences of the first degree
2.4. The Chinese remainder theorem
2.5. Polynomial congruences
2.6. Prime power moduli
2.7. Prime moduli
2.8. Primitive roots
2.9. Indices and binomial and exponential congruences
2.10. An application to cryptography
2.11. Pseudoprimes
Additional problems
3. Quadratic Residues
3.1. Introduction
3.2. Quadratic residues
3.3. The law of quadratic reciprocity
3.4. The Jacobi symbol
3.5. The Kronecker symbol
3.6. The solution of x^2 = D (mod m)
Additional problems
4. Approximation of Real Numbers
4.1. Introduction
4.2. Farey fractions
4.3. Approximation by rationals, I
4.4. Continued fractions
4.5. Periodic continued fractions
4.6. Approximation by rationals, II
4.7. Some factorization methods, II
4.8. Equivalent numbers
4.9. Decimal representation
Additional problems
5. Diophantine Equations, I
5.1. Introduction
5.2. Pythagorean triangles
5.3. Related quadratic equations
5.4. The equation ax^2 + by^2 + cz^2 = 0
Additional problems
6. Diophantine Equations, II
6.1. Introduction
6.2. Linear equations
6.3. Sums of two squares
6.4. Sums of four squares
6.5. Sums of other numbers of squares
6.6. Binary quadratic forms
6.7. The equation x^2 - D y^2 = N
6.8. A Pythagorean triangle problem
6.9. The equation ax^2 + bxy + cy^2 + dx + ey + f = 0
6.10. Waring's problem
Additional problems
7. Arithmetic Functions
7.1. Introduction
7.2. Dirichlet convolution
7.3. Multiplicative functions
7.4. Sum of divisors
7.5. Number of divisors
7.6. Euler's function
7.7. Characters
7.8. Trigonometrical sums
7.9. Additive functions
7.10. Linear recursion
Additional problems
8. The Average Order of Arithmetic Functions
8.1. Introduction
8.2. The greatest integer function
8.3. Preliminaries
8.4. Sum of divisors function
8.5. The number of divisors functions
8.6. Euler's function
8.7. Lattice point problems
Additional problems
9. Prime Number Theory
9.1. Introduction
9.2. Bertrand's postulate and Chebychev's theorem
9.3. The prime number theorem
9.4. Primes in arithmetic progressions
9.5. Order of magnitude of multiplicative functions
9.6. Summatory functions of additive functions
Additional problems
10. An Introduction To Algebraic Number Theory
10.1. Introduction
10.2. General considerations
10.3. Quadratic number fields
10.4. Applications to Diophantine equations
10.5. Concluding remarks
Additional problems
Tables
Bibliography
Index