Author(s): Benjamin Fine, Gerhard Rosenberger
Edition: 1
Publisher: Birkhäuser Boston
Year: 2006
Language: English
Pages: 354
Contents......Page 6
Preface......Page 9
1 Introduction and Historical Remarks......Page 13
2.1 The Ring of Integers......Page 19
2.2 Divisibility, Primes, and Composites......Page 23
2.3 The Fundamental Theorem of Arithmetic......Page 28
2.4 Congruences and Modular Arithmetic......Page 33
2.5 The Solution of Polynomial Congruences Modulo......Page 49
2.6 Quadratic Reciprocity......Page 57
EXERCISES......Page 63
3.1 The In.nitude of Primes......Page 67
3.2 Sums of Squares......Page 98
3.3 Dirichlet’s Theorem......Page 116
3.4 Twin Prime Conjecture and Related Ideas......Page 133
3.5 Primes Between x and 2x......Page 134
3.6 Arithmetic Functions and the Möbius Inversion Formula......Page 135
EXERCISES......Page 140
4.1 The Prime Number Theorem: Estimates and History......Page 145
4.2 Chebychev’s Estimate and Some Consequences......Page 148
4.3 Equivalent Formulations of the Prime Number Theorem......Page 161
4.4 The Riemann Zeta Function and the Riemann Hypothesis......Page 169
4.5 The Prime Number Theorem......Page 185
4.6 The Elementary Proof......Page 192
4.7 Some Extensions and Comments......Page 197
EXERCISES......Page 204
5.1 Primality Testing and Factorization......Page 209
5.2 Sieving Methods......Page 210
5.3 Primality Testing and Prime Records......Page 224
5.4 Cryptography and Primes......Page 246
5.5 The AKS Algorithm......Page 255
EXERCISES......Page 260
6.1 Algebraic Number Theory......Page 265
6.2 Unique Factorization Domains......Page 267
6.3 Algebraic Number Fields......Page 287
6.4 Algebraic Integers......Page 306
6.5 The Theory of Ideals......Page 323
EXERCISES......Page 339
Bibliography and Cited References......Page 345
Index......Page 349
