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

## Alan Baker: *A Concise Introduction to the Theory of Numbers*

Published $1984$, **Cambridge University Press**

- ISBN 0-521-28654-9.

### Subject Matter

### 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 $[x]$
- 2 Multiplicative functions
- 3 Euler's (totient) function $\phi(n)$
- 4 The Möbius function $\mu(n)$
- 5 The functions $\tau(n)$ and $\sigma(n)$
- 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