Elements of Number Theory

Description

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