1 THE THEORY OF DIVISIBILITY 1
1 Fundamental concepts and theorems 1
2 The greatest common divisor 2
3 The least common multiple 5
4 The Euclidean Algorithm and continued fractions 7
5 Prime numbers 11
6 Uniqueness of factorization into prime factors 12
Problems for Chapter 1 14
Numerical examples for Chapter 1 16
2 FUNDAMENTAL FUNCTIONS OF THE THEORY OF NUMBERS 17
1 Functions [x]，{x} 17
2 Summation over divisors of an integer 18
3 The Moebius function 19
4 Euler＇s function 20
Problems for Chapter 2 22
Numerical examples for Chapter 2 30
3 CONGRUENCES 31
1 Fundamental concepts 31
2 Properties of congruences similar to properties of equalities 32
2 Further properties of congruences 34
4 Complete system of residues 35
5 The reduced system of residues 36
6 Theorems of Euler and Fermat 37
Problems for Chapter 3 38
Numerical examples for Chapter 3 43
4 LINEAR CONGRUENCES 44
1 Fundamental concepts 44
2 Linear congruences 44
3 Simultaneous linear congruences 47
4 Congruences of any degree to a prime modulus 48
5 Congruences of any degree to a composite modulus 49
Problems for Chapter 4 52
Numerical examples for Chapter 4 56
5 QUADRATIC CONGRUENCES 58
1 General theorems 58
2 Legendre＇s symbol 59
3 Jacobi＇s symbol 64
4 The case of a composite modulus 67
Problems for Chapter 5 70
Numerical examples for Chapter 5 75
6 PRIMITIVE ROOTS AND INDICES 76
1 General theorems 76
2 Primitive roots to moduli pα and 2pα 76
3 Finding primitive roots to moduli pα and 2pα 78
4 Indices to moduli pα and 2pα 79
5 Applications of the theory of indices 81
6 Indices to modulus 2α 84
7 Indices to any composite modulus 86
Problems for Chapter 6 87
Numerical examples for Chapter 6 93
SOLUTIONS TO PROBLEMS 95
Solutions to Chapter 1 95
Solutions to Chapter 2 98
Solutions to Chapter 3 111
Solutions to Chapter 4 120
Solutions to Chapter 5 126
Solutions to Chapter 6 135
ANSWERS TO NUMERICAL EXAMPLES 145
Answers to Chapter 1 145
Answers to Chapter 2 145
Answers to Chapter 3 145
Answers to Chapter 4 145
Answers to Chapter 5 146
Answers to Chapter 6 146
Tables of Indices 148
Table of Odd Primes < 4000 and of their least primitive roots 154