Parshin A. N., Shafarevich I. R. (eds) Number Theory I. Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences, Springer)(1).pdf

文档介绍：A. N. Parshin I. R. Shafarevich (Eds.)
Number Theory I
Fundamental Problems,
Ideas and Theories
With 17 Figures
Springer
Encyclopaedia of
Mathematical Sciences
Volume 49
Editor-in-Chief: R. Y Gamkrelidze
Number Theory I
Introduction to Number Theory
Yu. I. Manin and A. A. Panchishkin
Contents
Preface to the English Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Part I. Problems and Tricks
Chapter 1. Elementary Number Theory ........................... 8
$1. Problems About Primes. Divisibility and Primality ........ 8
92. Diophantine Equations of Degree One and Two ........... 20
§3. Cubic Diophantine Equations ........................... 34
94. The Structure of the Continuum. Approximations and
Continued Fractions ................................... 44
Chapter 2. Some Modern Problems of Elementary Number Theory . . . 49
31. Factorization and Public Key Cryptosystems . . . . . . . . . . . . . . 49
§2. Deterministic Primality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
§3. Factorization of Large Integers . . . . . . . . . . . . . . . . . . . . . . . . . . 61
54. Diophantine Approximation and the Irrationality of c(3) . . . 70
Part II. Ideas and Theories
Chapter 1. Induction and Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
$1. Elementary Number Theory From the Point of View of
Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2 Contents
$2. Diophantine Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
$3. Partially Recursive Functions and Ennumerable Sets . . . 83
$4. Diophantineness of a Set and Algorithmic Undecidability 92
Chapter 2. Arithmetic of Algebraic Numbers ................... . . 94
$1. Algeb