Preface to the second edition
Introduction
1. The fundamental theorem of arithmetic
Division algorithm
Greatest common divisor and Euclidean algorithm
Unique factorisation into primes
Infinity of primes
Mersenne primes
Summary
Historical note
Notes and answers
2. Modular addition and Euler’s φ function
Congruence classes and the Chinese remainder theorem
The groups (Zn, +) and their generators
Euler’s φ function
Summing Euler’s function over divisors
Summary
Historical note
Notes and answers
3. Modular multiplication
Fermat’s theorem
Wilson’s theorem
Linear congruences
Fermat-Euler theorem
Simultaneous linear congruences
Lagrange’s theorem for polynomials
Primitive roots
Chevalley’s theorem
RSA codes
Summary
Historical note
Notes and answers
4. Quadratic residues
Quadratic residues and the Legendre symbol
Gauss’ lemma
Law of quadratic reciprocity
Summary
Historical note
Notes and answers
5. The equation xn + yn = zn, for n = 2, 3, 4
The equation x2 + y2 = z2
The equation x4 + y4 = z4
The equation x2 + y2 + z2 = t2
The equation x3 + y3 = z3
Historical note
Notes and answers
6. Sums of squares
Sums of two squares
Sums of four squares
Sums of three squares
Triangular numbers
Historical note
Notes and answers
7. Partitions
Ferrers’ graphs
Generating functions
Euler’s theorem
Summary
Historical note
Notes and answers
8. Quadratic forms
Unimodular transformations
Equivalent quadratic forms
Discriminant
Proper representation
Reduced forms
Automorphs of definite quadratic forms
Summary
Historical note
Notes and answers
9. Geometry of numbers
Subgroups of a square lattice
Minkowski’s theorem in two dimensions
Subgroups of a cubic lattice
Minkowski’s theorem in three dimensions
Legendre’s theorem on ax2 + by2 + cz2 = 0
Summary
Historical note
Notes and answers
10. Continued fractions
Irrational square roots
Convergence
Purely periodic continued fractions
Pell’s equation
Lagrange’s theorem on quadratic irrationals
Automorphs of the indefinite form ax2 – by2
Summary
Historical note
Notes and answers
11. Approximation of irrationals by rationals
Naive approach
Farey sequences
Hurwitz’ theorem
Liouville’s theorem
Summary
Historical note
Notes and answers
Bibliography
Index