Number Theory

Table of contents :
Cover
Title page
Copyright
Contents
Introduction
Structure of the book
Exercises
Short overview of the individual chapters
Technical details
Commemoration
Acknowledgements
Chapter 1. Basic Notions
1.1. Divisibility
Exercises 1.1
1.2. Division Algorithm
Exercises 1.2
1.3. Greatest Common Divisor
Exercises 1.3
1.4. Irreducible and Prime Numbers
Exercises 1.4
1.5. The Fundamental Theorem of Arithmetic
Exercises 1.5
1.6. Standard Form
Exercises 1.6
Chapter 2. Congruences
2.1. Elementary Properties
Exercises 2.1
2.2. Residue Systems and Residue Classes
Exercises 2.2
2.3. Euler’s Function ?
Exercises 2.3
2.4. The Euler–Fermat Theorem
Exercises 2.4
2.5. Linear Congruences
Exercises 2.5
2.6. Simultaneous Systems of Congruences
Exercises 2.6
2.7. Wilson’s Theorem
Exercises 2.7
2.8. Operations with Residue Classes
Exercises 2.8
Chapter 3. Congruences of Higher Degree
3.1. Number of Solutions and Reduction
Exercises 3.1
3.2. Order
Exercises 3.2
3.3. Primitive Roots
Exercises 3.3
3.4. Discrete Logarithm (Index)
Exercises 3.4
3.5. Binomial Congruences
Exercises 3.5
3.6. Chevalley’s Theorem, Kőnig–Rados Theorem
Exercises 3.6
3.7. Congruences with Prime Power Moduli
Exercises 3.7
Chapter 4. Legendre and Jacobi Symbols
4.1. Quadratic Congruences
Exercises 4.1
4.2. Quadratic Reciprocity
Exercises 4.2
4.3. Jacobi Symbol
Exercises 4.3
Chapter 5. Prime Numbers
5.1. Classical Problems
Exercises 5.1
5.2. Fermat and Mersenne Primes
Exercises 5.2
5.3. Primes in Arithmetic Progressions
Exercises 5.3
5.4. How Big Is ?(?)?
Exercises 5.4
5.5. Gaps between Consecutive Primes
Exercises 5.5
5.6. The Sum of Reciprocals of Primes
Exercises 5.6
5.7. Primality Tests
Exercises 5.7
5.8. Cryptography
Exercises 5.8
Chapter 6. Arithmetic Functions
6.1. Multiplicative and Additive Functions
Exercises 6.1
6.2. Some Important Functions
Exercises 6.2
6.3. Perfect Numbers
Exercises 6.3
6.4. Behavior of ?(?)
Exercises 6.4
6.5. Summation and Inversion Functions
Exercises 6.5
6.6. Convolution
Exercises 6.6
6.7. Mean Value
Exercises 6.7
6.8. Characterization of Additive Functions
Exercises 6.8
Chapter 7. Diophantine Equations
7.1. Linear Diophantine Equation
Exercises 7.1
7.2. Pythagorean Triples
Exercises 7.2
7.3. Some Elementary Methods
Exercises 7.3
7.4. Gaussian Integers
Exercises 7.4
7.5. Sums of Squares
Exercises 7.5
7.6. Waring’s Problem
Exercises 7.6
7.7. Fermat’s Last Theorem
Exercises 7.7
7.8. Pell’s Equation
Exercises 7.8
7.9. Partitions
Exercises 7.9
Chapter 8. Diophantine Approximation
8.1. Approximation of Irrational Numbers
Exercises 8.1
8.2. Minkowski’s Theorem
Exercises 8.2
8.3. Continued Fractions
Exercises 8.3
8.4. Distribution of Fractional Parts
Exercises 8.4
Chapter 9. Algebraic and Transcendental Numbers
9.1. Algebraic Numbers
Exercises 9.1
9.2. Minimal Polynomial and Degree
Exercises 9.2
9.3. Operations with Algebraic Numbers
Exercises 9.3
9.4. Approximation of Algebraic Numbers
Exercises 9.4
9.5. Transcendence of ?
Exercises 9.5
9.6. Algebraic Integers
Exercises 9.6
Chapter 10. Algebraic Number Fields
10.1. Field Extensions
Exercises 10.1
10.2. Simple Algebraic Extensions
Exercises 10.2
10.3. Quadratic Fields
Exercises 10.3
10.4. Norm
Exercises 10.4
10.5. Integral Basis
Exercises 10.5
Chapter 11. Ideals
11.1. Ideals and Factor Rings
Exercises 11.1
11.2. Elementary Connections to Number Theory
Exercises 11.2
11.3. Unique Factorization, Principal Ideal Domains, and Euclidean Rings
Exercises 11.3
11.4. Divisibility of Ideals
Exercises 11.4
11.5. Dedekind Rings
Exercises 11.5
11.6. Class Number
Exercises 11.6
Chapter 12. Combinatorial Number Theory
12.1. All Sums Are Distinct
Exercises 12.1
12.2. Sidon Sets
Exercises 12.2
12.3. Sumsets
Exercises 12.3
12.4. Schur’s Theorem
Exercises 12.4
12.5. Covering Congruences
Exercises 12.5
12.6. Additive Complements
Exercises 12.6
Answers and Hints
A.1. Basic Notions
A.2. Congruences
A.3. Congruences of Higher Degree
A.4. Legendre and Jacobi Symbols
A.5. Prime Numbers
A.6. Arithmetic Functions
A.7. Diophantine Equations
A.8. Diophantine Approximation
A.9. Algebraic and Transcendental Numbers
A.10. Algebraic Number Fields
A.11. Ideals
A.12. Combinatorial Number Theory
Historical Notes
Tables
Primes 2–1733
Primes 1741–3907
Prime Factorization
Mersenne Numbers
Fermat Numbers
Index
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