Preface 1 Introduction 1.1 Diophantine Equations 1.2 Modular Arithmetic 1.3 Primes and the Distribution of Primes 1.4 Cryptography 2 Divisibility 2.1 Divisibility 2.2 Euclid's Theorem 2.3 Euclid's Original Proof 2.4 The Sieve of Eratosthenes 2.5 The Division Algorithm 2.5.1 A Cryptographic Application 2.6 The Greatest Common Divisor 2.7 The Euclidean Algorithm 2.7.1 The Extended Euclidean Algorithm 2.8 Other Bases 2.9 Fermat and Mersenne Numbers 2.10 Chapter Highlights 2.11 Problems 2.11.1 Exercises 2.11.2 Projects 2.11.3 Computer Explorations 2.11.4 Answers to "Check Your Understanding" 3 Linear Diophantine Equations 3.1 ax + by=c 3.2 The Postage Stamp Problem 3.3 Chapter Highlights 3.4 Problems 3.4.1 Exercises 3.4.2 Answers to "Check Your Understanding" 4 Unique Factorization 4.1 The Starting Point 4.2 The Fundamental Theorem of Arithmetic 4.3 Euclid and the Fundamental Theorem of Arithmetic 4.4 Chapter Highlights 4.5 Problems 4.5.1 Exercises 4.5.2 Projects 4.5.3 Answers to "Check Your Understanding" 5 Applications of Unique Factorization 5.1 A Puzzle 5.2 Irrationality Proofs 5.2.1 Four More Proofs That √2 Is Irrational 5.3 The Rational Root Theorem 5.4 Pythagorean Triples 5.5 Differences of Squares 5.6 Prime Factorization of Factorials 5.7 The Riemann Zeta Function 5.7.1 ?1/p Diverges
5.8 Chapter Highlights 5.9 Problems 5.8.1 Exercises 5.9.2 Projects 5.9.3 Computer Explorations …… 6 Congruences 7 Classical Cryptosystems 8 Fermat, Euler, and Wilson 9 RSA 10 Polynomial Congruences 11 Order and Primitive Roots 12 More Cryptographic Applications 13 Quadratic Reciprocity 14 Primality and Factorization 15 Geometry of Numbers 16 Arithmetic Functions 17 Continued Fractions 18 Gaussian Integers 19 Algebraic Integers 20 The Distribution of Primes 21 Epilogue: Fermat's Last Theorem A Supplementary Topics B Answers and Hints for Odd-Numbered Exercises Index 編輯手記
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