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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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Thoroughly Revised And Updated, The New Second Edition Of Neville Robbins’ Beginning Number Theory Includes All Of The Major Topics Covered In A Classic Number Theory Course And Blends In Numerous Applications And Specialized Treatments Of Number Theory, Including Cryptology, Fibonacci Numbers, And Computational Number Theory. The Text Strikes A Balance Between Traditional And Algorithmic Approaches To Elementary Number Theory And Is Supported With Numerous Exercises, Applications, And Case Studies Throughout. Computer Exercises For CAS Systems Are Also Included.

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