Table of Contents
Preface v
Structure of the book viii
Chapter 1 Irrationality and diophantine approximation
1.1 Irrationality of $$$ 1
1.2 Irrationality of e 2
1.3 Irrationality of π 3
1.4 Irrationality of the values of the Tschakaloff function 3
1.5 Diophantine approximation 5
1.6 Methodological remarks 8
Exercises 8
Chapter 2 Representations of real numbers by infinite series and products
2.1 p-adic expansion of a real number 11
2.2 Expansion of a real number in Engel series 14
2.3 Cantor infinite product 15
Exercises 17
Chapter 3 Continued fractions
3.1 Introduction 21
3.2 A criterion of convergence 24
3.3 How to divide by zero 27
3.4 Expansion in continued fraction in Σ 28
3.5 A quotient of Bessel functions 30
3.6 Continued fractions and irrationality 32
Exercises 35
Chapter 4 Regular continued fractions
4.1 Regular continued fraction expansion of a positive real number 39
4.2 The regular continued fraction expansion of e 43
4.3 The diophantine equation ax + by = c 44
4.4 Regular continued fractions and diophantine approximation 45
4.5 Quadratic irrational numbers and continued fractions 47
4.6 Pell's equation 50
Exercises 53
Chapter 5 Quadratic fields and diophantine equations
5.1 Quadratic fields 57
5.2 Ring of integers of a quadratic field 58
5.3 Units of the ring of integers of a quadratic field 60
5.4 Unique factorization theorem in Z 63
5.5 Primes and irreducibles 66
5.6 Euclidean domains 68
5.7 Diophantine equations 70
Exercises 73
Chapter 6 Squares and sums of squares
6.1 Sums of two squares 77
6.2 Finite algebraic structures 79
6.3 Legendre symbol 81
6.4 Computations in $$$ 86
6.5 Binary quadratic forms with integer coefficients 88
6.6 Sums of four squares 93
Exercises 95
Chapter 7 Arithmetical functions
7.1 Ordinary generating function 99
7.2 Lambert series 102
7.3 Jacobi's triple product identity 104
7.4 Sums of two squares 106
7.5 Jacobi's theorem on sums of four squares 108
7.6 Euler function φ(n) 108
7.7 Average order of r2(n) 110
7.8 The series of primes: the function π(n) 111
Exercises 115
Chapter 8 Padé approximants
8.1 Introduction 121
8.2 Gauss hypergeometric function and Padé approximants of the binomial function 122
8.3 Confluent hypergeometric functions and Padé approximants of the exponential function 125
8.4 Arithmetical applications 126
Exercises 129
Chapter 9 Algebraic numbers and irrationality measures
9.1 Algebraic numbers 134
9.2 Algebraic integers 136
9.3 Transcendental numbers and Liouville's theorem 139
9.4 Irrationality measures 141
9.5 Diophantine equations and irrationality measures 144
9.6 The theorems of Thue and Roth 147
Exercises 148
Chapter 10 Number fields
10.1 Algebraic number fields 153
10.2 Conjugates, norms and traces 156
10.3 Ring of integers of a number field 158
10.4 Units 160
10.5 Discriminants and integral basis 163
10.6 Fermat's equation x5 + y5 = z5 167
Exercises 170
Chapter 11 Ideals
11.1 Fractional ideals and ideals of a number field 174
11.2 Arithmetic in J(AK) 176
11.3 Norm of an ideal 178
11.4 Factorization of (p), p prime, as a product of prime ideals 182
11.5 Class group and class number 185
11.6 Application to Mordell's equation y2 = x3 + k 188
Exercises 190
Chapter 12 Introduction to transcendence methods
12.1 Algebraic and transcendental functions 195
12.2 House of an algebraic integer 197
12.3 Mahler's transcendence method (1929) 198
12.4 Methodological remarks and Siegel's lemma 200
12.5 Hermite-Lindemann theorem 202
12.6 Gelfond-Schneider theorem 204
12.7 Siegel-Shidlovski method 207
Exercises 208
Solutions to the exercises
Chapter 1 213
Chapter 2 220
Chapter 3 225
Chapter 4 233
Chapter 5 244
Chapter 6 255
Chapter 7 265
Chapter 8 277
Chapter 9 289
Chapter 10 301
Chapter 11 310
Chapter 12 320
Bibliography 331
Index 333
