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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