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

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 $$[x]$$
 * 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