Book:Alan Baker/A Concise Introduction to the Theory of Numbers

Alan Baker: A Concise Introduction to the Theory of Numbers

Published $\text {1984}$, Cambridge University Press

ISBN 0-521-28654-9

Subject Matter

• Number Theory

Contents

Preface

Introduction: Gauss and number theory

1 Divisibility

1 Foundations

2 Division algorithm

3 Greatest common divisor

4 Euclid's algorithm

5 Fundamental theorem

6 Properties of the primes

7 Further reading

8 Exercises

2 Arithmetical functions

1 The function $\sqbrk x$

2 Multiplicative functions

3 Euler's (totient) function $\map \phi n$

4 The Möbius function $\map \mu n$

5 The functions $\map \tau n$ and $\map \sigma n$

6 Average orders

7 Perfect numbers

8 The Riemann zeta-function

9 Further reading

10 Exercises

3 Congruences

1 Definitions

2 Chinese remainder theorem

3 The theorems of Fermat and Euler

4 Wilson's theorem

5 Lagrange's theorem

6 Primitive roots

7 Indices

8 Further reading

9 Exercises

4 Quadratic residues

1 Legendre's symbol

2 Euler's criterion

3 Gauss' lemma

4 Law of quadratic reciprocity

5 Jacobi's symbol

6 Further reading

7 Exercises

5 Quadratic forms

1 Equivalence

2 Reduction

3 Representations by binary forms

4 Sums of two squares

5 Sums of four squares

6 Further reading

7 Exercises

6 Diophantine approximation

1 Dirichlet's theorem

2 Continued fractions

3 Rational approximations

4 Quadratic irrationals

5 Liouville's theorem

6 Transcendental numbers

7 Minkowski's theorem

8 Further reading

9 Exercises

7 Quadratic fields

1 Algebraic number fields

2 The quadratic field

3 Units

4 Primes and factorization

5 Euclidean fields

6 The Gaussian field

7 Further reading

8 Exercises

8 Diophantine equations

1 The Pell equation

2 The Thue equation

3 The Mordell equation

4 The Fermat equation

5 The Catalan equation

6 Further reading

7 Exercises