An introduction to the theory of numbers

Title page......Page 1
Date-line......Page 2
CONTENTS......Page 3
1 Fundamental concepts and theorems......Page 5
2 The greatest common divisor......Page 6
3 The least common multiple......Page 9
4 The Euclidean Algorithm and continued fractions......Page 11
5 Prime numbers......Page 15
6 Uniqueness of factorization into prime factors......Page 16
Problems for Chapter 1......Page 18
Numerical examples for Chapter 1......Page 20
1 Functions [x], {x}......Page 21
2 Summation over divisors of an integer......Page 22
3 The Moebius function......Page 23
4 Euler's function......Page 24
Problems for Chapter 2......Page 26
Numerical examples for Chapter 2......Page 34
1 Fundamental concepts......Page 35
2 Properties of congruences similar to properties of equalities......Page 36
3 Further properties of congruences......Page 38
4 Complete system of residues......Page 39
5 The reduced system of residues......Page 40
6 Theorems of Euler and Fermat......Page 41
Problems for Chapter 3......Page 42
Numerical examples for Chapter 3......Page 47
2 Linear congruences......Page 48
3 Simultaneous linear congruences......Page 51
4 Congruences of any degree to a prime modulus......Page 52
5 Congruences of any degree to a composite modulus......Page 53
Problems for Chapter 4......Page 56
Numerical examples for Chapter 4......Page 60
1 General theorems......Page 62
2 Legendre's symbol......Page 63
3 Jacobi's symbol......Page 68
4 The case of a composite modulus......Page 71
Problems for Chapter 5......Page 74
Numerical examples for Chapter 5......Page 79
2 Primitive roots to moduli $p^\alpha$ and 2$p^\alpha$......Page 80
3 Finding primitive roots to moduli $p^\alpha$ and 2$p^\alpha$......Page 82
4 Indices to moduli $p^\alpha$ and 2$p^\alpha$......Page 83
5 Applications of the theory of indices......Page 85
6 Indices to modulus $2\alpha$......Page 88
7 Indices to any composite modulus......Page 90
Problems for Chapter 6......Page 91
Numerical examples for Chapter 6......Page 97
Solutions to Chapter 1......Page 99
Solutions to Chapter 2......Page 102
Solutions to Chapter 3......Page 115
Solutions to Chapter 4......Page 124
Solutions to Chapter 5......Page 130
Solutions to Chapter 6......Page 139
Answers to Chapter 4......Page 149
Answers to Chapter 6......Page 150
Tables of Indices......Page 152
Table of Odd Primes < 4000 and of their least primitive roots......Page 158