Elements of Number Theory

chapter 1
PRELIMINARY
CONSIDERATIONS 1
1.1 ORDERED INTEGRAL DOMAINS 1
1.2 MATHEMATICAL INDUCTION 6
1.3 NUMBER BASES 9
1.4 NOTATIONS FOR SUMS AND PRODUCTS 15
chapter 2
DIVISIBILITY
PROPERTIES
OF INTEGERS
23
2.1
PRIME AND COMPOSITE NUMBERS
23
2.2
SOME PROPERTIES OF PRIME NUMBERS
27
2.3
THE GREATEST COMMON DIVISOR
34
2.4
THE GENERALIZED
GREATEST COMMON DIVISOR
40
2.5
PROPERTIES
OF THE GREATEST COMMON DIVISOR
43
2.6
LINEAR DIOPHANTINE EQUATIONS
46
2.7
THE FUNDAMENTAL THEOREM
OF ARITHMETIC
51
2.8
SUM OF DIVISORS
58
2.9
NUMBER OF DIVISORS
63
2.10
PERFECT NUMBERS
65
2.11
MERSENNE NUMBERS
69
2.12
THE EULER -FUNCTION
72
2.13
PROPERTIES OF THE EULER -FUNCTION
78
2.14
SOLUTION OF THE EQUATION (j)(x) = m
81
chapter 3
THE THEORY OF
CONGRUENCES 87
3.1 DEFINITIONS
AND ELEMENTAR Y PROPERTIES 87
3.2 SOME CONGRUENCE THEOREMS 92
3.3 AN APPLICATION
OF THE CONGR UENCE RELA TION 99
3.4 REDUCED RESIDUE SYSTEMS MODULO m 103
3.5 THE THEOREMS OF EULER AND FERMAT 108
3.6 LINEAR CONGRUENCES 113
3.7 LINEAR CONGRUENCES
AND LINE A R DIOPHA NTINE EQ UA TIONS 120
3.8 WILSON\'S THEOREM 124
3.9 LINEAR CONGRUENCES IN TWO VARIABLES 129
3.10 THE CHINESE REMAINDER THEOREM 135
3.11 QUADRATIC CONGRUENCES 143
chapter 4
CONTINUED
FRACTIONS 149
4.1 FINITE CONTINUED FRACTIONS 149
4.2 INFINITE CONTINUED FRACTIONS 156
4.3 CONVERGENTS 162
4.4 EVALUATION OF CONVERGENTS 167
4.5 CONVERGENTS AS DETERMINANTS 172
4.6 SOME PROPERTIES OF CONVERGENTS 177
4.7 CONTINUED FRACTIONS
A ND LINE A R DIOPHA NTINE EQUA TIONS 180
4.8 THEOREMS ON
INFINITE CONTINUED FRACTIONS 182
4.9 APPROXIMATION THEOREMS 187
4.10 USE OF CONTINUED FRA CTIONS
IN SOL VING Q UA DRA TIC EQ UA TIONS 192
4.11 PERIODIC CONTINUED FRACTIONS 196
4.12 CONTINUED FRACTIONS FOR Qk 202
ANSWERS
TO SELECTED
EXERCISES 207
LIST OF
PRIME NUMBERS
LESS THAN
10,000 238
INDEX
242