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목차
Preface ix
About This Book ix
Features x
Preliminaries 1
Introduction 1
Conjectures, Theorems, and Proofs 4
Well-Ordering and Induction 8
Well-Ordering Principle 8
Sigma Notation and Product Notation 16
Binomial Coefficients 18
Greatest Integer Function 23
Review Exercises 26
Divisibility 27
Introduction 27
Divisibility, Greatest Common Divisor, Euclid's Algorithm 28
Greatest Common Divisor via Euclid's Algorithm 31
Least Common Multiple 40
Representations of Integers 43
Decimal Representations of Integers 43
Binary Representations of Integers 46
Review Exercises 50
Primes 53
Introduction 53
Primes, Prime Counting Function, Prime Number Theorem 54
Test of Primality by Trial Division 56
Sieve of Eratosthenes, Canonical Factorization, Fundamental Theorem of Arithmetic 61
Sieve of Eratosthenes 61
Determining the Canonical Factorization of a Natural Number 63
Review Exercises 70
Congruences 71
Introduction 71
Congruences and Equivalence Relations 72
Equivalence Relations 73
Linear Congruences 79
Linear Diophantine Equations and the Chinese Remainder Theorem 89
Polynomial Congruences 94
Modular Arithmetic: Fermat's Theorem 99
Wilson's Theorem and Fermat Numbers 106
Pythagorean Equation 111
Review Exercises 116
Arithmetic Functions 119
Introduction 119
Sigma Function, Tau Function, Dirichlet Product 119
Dirichlet Inverse, Moebius Function, Euler's Function, Euler's Theorem 134
An Application to Algebra 143
Review Exercises 146
Primitive Roots and Indices 147
Introduction 147
Primitive Roots: Definition and Properties 148
Primitive Roots: Existence 155
Indices 164
Review Exercises 167
Quadratic Congruences 169
Introduction 169
Quadratic Residues and the Legendre Symbol 170
Gauss' Lemma and the Law of Quadratic Reciprocity 175
Solution of Quadratic Congruences 186
Algorithm for Solving Quadratic Congruences 187
Quadratic Congruences with Composite Moduli 190
Jacobi Symbol 194
Review Exercises 198
Sums of Squares 201
Introduction 201
Sums of Two Squares 201
Sums of Four Squares 210
Review Exercises 218
Continued Fractions 221
Introduction 221
Finite Continued Fractions 221
Infinite Continued Fractions 233
Approximation by Continued Fractions 239
Periodic Continued Fractions: I 246
Periodic Continued Fractions: II 253
Review Exercises 261
Nonlinear Diophantine Equations 263
Introduction 263
Fermat's Last Theorem 262
Pell's Equation: x2 -- Dy2 = 1 264
Mordell's Equation: x3 = y2 + k 276
Review Exercises 278
Computational Number Theory 279
Introduction 279
Pseudoprimes and Carmichael Numbers 283
Miller's Test and Strong Pseudoprimes 288
Factoring: Fermat's Method and the Continued Fraction Method 292
Trial Division 292
Fermat's Method 292
Continued Fraction Method 292
Quadratic Sieve Method 296
Pollard p -- 1 Method 299
Review Exercises 301
Cryptology 303
Introduction 303
Character Ciphers 305
Block Ciphers 308
One-Time Pads: Exponential Ciphers 309
Public-Key Cryptography 311
Signatures 312
Review Exercises 313
Appendix A Some Open Questions in Elementary Number Theory 315
Appendix B Tables 321
Appendix C Answers to Selected Exercises 323
Bibliography 333
Index 335
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