Elementary number theory / David M. Burton

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Contents

Preface

New to This Edition

1 Preliminaries

1.1 Mathematical Induction

1.2 The Binomial Theorem

2 Divisibility Theory in the Integers

2.1 Early Number Theory

2.2 The Division Algorithm

2.3 The Greatest Common Divisor

2.4 The Euclidean Algorithm

2.5 The Diophantine Equation ax + by = c

3 Primes and Their Distribution

3.1 The Fundamental Theorem of Arithmetic

3.2 The Sieve of Eratosthenes

3.3 The Goldbach Conjecture

4 The Theory of Congruences

4.1 Carl Friedrich Gauss

4.2 Basic Properties of Congruence

4.3 Binary and Decimal Representations of Integers

4.4 Linear Congruences and the Chinese Remainder Theorem

5 Fermat's Theorem nominariqnich will add

5.1 Pierre de Fermat

5.2 Fermat's Little Theorem and Pseudoprimes

5.3 Wilson's Theorem

5.4 The Fermat-Kraitchik Factorization Method

6 Number-Theoretic Functions

6.1 The Sum and Number of Divisors

6.2 The Möbius Inversion Formula

6.3 The Greatest Integer Function

6.4 An Application to the Calendar

7 Euler's Generalization of Fermat's Theorem

7.1 Leonhard Euler

7.2 Euler's Phi-Function

7.3 Euler's Theorem

7.4 Some Properties of the Phi-Function

8 Primitive Roots and Indices

8.1 The Order of an Integer Modulo n

8.2 Primitive Roots for Primes

8.3 Composite Numbers Having Primitive Roots

8.4 The Theory of Indices

9 The Quadratic Reciprocity Law

9.1 Euler's Criterion

9.2 The Legendre Symbol and Its Properties

9.3 Quadratic Reciprocity

9.4 Quadratic Congruences with Composite Moduli

10 Introduction to Cryptography

10.1 From Caesar Cipher to Public Key Cryptography

10.2 The Knapsack Cryptosystem

10.3 An Application of Primitive Roots to Cryptography

11 Numbers of Special Form

11.1 Marin Mersenne

11.2 Perfect Numbers

11.3 Mersenne Primes and Amicable Numbers

11.4 Fermat Numbers

12 Certain Nonlinear Diophantine Equations

12.1 The Equation x ^ 2 + y ^ 2 = z ^ 2

12.2 Fermat's Last Theorem

13 Representation of Integers as Sums of Squares

13.1 Joseph Louis Lagrange

13.2 Sums of Two Squares

13.3 Sums of More Than Two Squares

14 Fibonacci Numbers

14.1 Fibonacci

14.2 The Fibonacci Sequence

14.3 Certain Identities Involving Fibonacci Numbers

15 Continued Fractions

15.1 Srinivasa Ramanujan

15.2 Finite Continued Fractions

15.3 Infinite Continued Fractions

15.4 Farey Fractions

15.5 Pell's Equation

16 Some Modern Developments

16.1 Hardy, Dickson, and Erdös

16.2 Primality Testing and Factorization

16.3 An Application to Factoring: Remote Coin Flipping

16.4 The Prime Number Theorem and Zeta Function

Miscellaneous Problems

Appendixes

General References

Suggested Further Reading

Tables

Answers to Selected Problems

Index