Elementary number theory

Description

Cover Preface What Is Number Theory? 1. The Integers 1.1. Numbers and Sequences 1.2. Sums and Products 1.3. Mathematical Induction 1.4. The Fibonacci Numbers 1.5. Divisibility 2. Integer Representations and Operations 2.1. Representations of Integers 2.2. Computer Operations with Integers 2.3. Complexity of Integer Operations 3. Primes and Greatest Common Divisors 3.1. Prime Numbers 3.2. The Distribution of Primes 3.3. Greatest Common Divisors and their Properties 3.4. The Euclidean Algorithm 3.5. The Fundamental Theorem of Arithmetic 3.6. Factorization Methods and the Fermat Numbers 3.7. Linear Diophantine Equations 4. Congruences 4.1. Introduction to Congruences 4.2. Linear Congruences 4.3. The Chinese Remainder Theorem 4.4. Solving Polynomial Congruences 4.5. Systems of Linear Congruences 4.6. Factoring Using the Pollard Rho Method 5. Applications of Congruences 5.1. Divisibility Tests 5.2. The Perpetual Calendar 5.3. Round-Robin Tournaments 5.4. Hashing Functions 5.5. Check Digits 6. Some Special Congruences 6.1. Wilson’s Theorem and Fermat’s Little Theorem 6.2. Pseudoprimes 6.3. Euler’s Theorem 7. Multiplicative Functions 7.1. The Euler Phi-Function 7.2. The Sum and Number of Divisors 7.3. Perfect Numbers and Mersenne Primes 7.4. Möbius Inversion 7.5. Partitions 8. Cryptology 8.1. Character Ciphers 8.2. Block and Stream Ciphers 8.3. Exponentiation Ciphers 8.4. Public Key Cryptography 8.5. Knapsack Ciphers 8.6. Cryptographic Protocols and Applications 9. Primitive Roots 9.1. The Order of an Integer and Primitive Roots 9.2. Primitive Roots for Primes 9.3. The Existence of Primitive Roots 9.4. Discrete Logarithms and Index Arithmetic 9.5. Primality Tests Using Orders of Integers and Primitive Roots 9.6. Universal Exponents 10. Applications of Primitive Roots and the Order of an Integer 10.1. Pseudorandom Numbers 10.2. The ElGamal Cryptosystem 10.3. An Application to the Splicing of Telephone Cables 11. Quadratic Residues 11.1. Quadratic Residues and Nonresidues 11.2. The Law of Quadratic Reciprocity 11.3. The Jacobi Symbol 11.4. Euler Pseudoprimes 11.5. Zero-Knowledge Proofs 12. Decimal Fractions and Continued Fractions 12.1. Decimal Fractions 12.2. Finite Continued Fractions 12.3. Infinite Continued Fractions 12.4. Periodic Continued Fractions 12.5. Factoring Using Continued Fractions 13. Some Nonlinear Diophantine Equations 13.1. Pythagorean Triples 13.2. Fermat’s Last Theorem 13.3. Sums of Squares 13.4. Pell’s Equation 13.5. Congruent Numbers 14. The Gaussian Integers 14.1. Gaussian Integers and Gaussian Primes 14.2. Greatest Common Divisors and Unique Factorization 14.3. Gaussian Integers and Sums of Squares Appendix A. Axioms for the Set of Integers Appendix B. Binomial Coefficients Appendix C. Using Maple and Mathematica for Number Theory C.1. Using Maple for Number Theory C.2. Using Mathematica for Number Theory Appendix D. Number Theory Web Links Appendix E. Tables Answers to Odd-Numbered Exercises Bibliography Index of Biographies Index Photo Credits List of Symbols