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

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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