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Erica Flapan joined the mathematics department at Pomona College in 1986. She has taught a wide range of mathematics courses and has numerous publications in both 3-dimensional topology and applications of topology to chemistry. In addition to her research and teaching in mathematics, she is interested in improving the mathematical background of science students. She developed a course entitled "Problem Solving in the Sciences,” which aims to teach students the mathematics they need in order to succeed in science and economics.
Tim Marks is a Research Scientist at Mitsubishi Electric Research Laboratories in Cambridge, Massachusetts. After teaching high school mathematics and physics for three years in Glenview, Illinois, he worked for three years as a mathematics textbook editor at McDougal Littell/ Houghton Mifflin. Marks and Pommersheim have taught number theory at the Johns Hopkins University's Center for Talented Youth (CTY) summer program for 18 years.
| Preface | p. viii |
| Structure of the Text | p. ix |
| To the Student | p. x |
| To the Instructor | p. xi |
| Acknowledgements | p. xiv |
| Prologue: Number Theory Through the Ages | p. xvi |
| Numbers, Rational and Irrational (Historical figures: Pythagoras and Hypatia) | p. 6 |
| Numbers and the Greeks | p. 6 |
| Numbers You Know | p. 13 |
| A First Look at Proofs | p. 17 |
| Irrationality of ?2 | p. 28 |
| Using Quantifiers | p. 32 |
| Mathematical Induction (Historical figure: Noether) | p. 42 |
| The Principle of Mathematical Induction | p. 42 |
| Strong Induction and the Well-Ordering Principle | p. 55 |
| The Fibonacci Sequence and the Golden Ratio | p. 67 |
| The Legend of the Golden Ratio | p. 76 |
| Divisibility and Primes (Historical figure: Eratosthenes) | p. 92 |
| Basic Properties of Divisibility | p. 92 |
| Prime and Composite Numbers | p. 98 |
| Patterns in the Primes | p. 104 |
| Common Divisors and Common Multiples | p. 116 |
| The Division Theorem | p. 124 |
| Applications of god and 1cm | p. 138 |
| The Euclidean Algorithm (Historical figure: Euclid) | p. 148 |
| The Euclidean Algorithm | p. 148 |
| Finding the Greatest Common Divisor | p. 156 |
| A Greeker Argument that ?2 Is Irrational | p. 172 |
| Linear Diophantine Equations (Historical figure: Diophantus) | p. 182 |
| The Equation aX + bY= 1 | p. 182 |
| Using the Euclidean Algorithm to Find a Solution | p. 191 |
| The Diophantine Equation aX + bY = n | p. 200 |
| Finding All Solutions to a Linear Diophantine Equation | p. 205 |
| The Fundamental Theorem of Arithmetic (Historical figure: Mersenne) | p. 216 |
| The Fundamental Theorem | p. 216 |
| Consequences of the Fundamental Theorem | p. 225 |
| Modular Arithmetic (Historical figure: Gauss) | p. 241 |
| Congruence Modulo n | p. 241 |
| Arithmetic with Congruences | p. 254 |
| Check-Digit Schemes | p. 267 |
| The Chinese Remainder Theorem | p. 274 |
| The Gregorian Calendar | p. 288 |
| The Mayan Calendar | p. 296 |
| Modular Number Systems (Historical figure: Turing) | p. 307 |
| The Number System Zn: An Informal View | p. 307 |
| The Number System Zn: Definition and Basic Properties | p. 310 |
| Multiplicative Inverses in Zn | p. 322 |
| Elementary Cryptography | p. 338 |
| Encryption Using Modular Multiplication | p. 343 |
| Exponents Modulo n (Historical figure: Fermat) | p. 355 |
| Fermat's Little Theorem | p. 355 |
| Reduced Residues and the Euler ?-Function | p. 368 |
| Euler's Theorem | p. 379 |
| Exponentiation Ciphers with a Prime Modulus | p. 390 |
| The RSA Encryption Algorithm | p. 399 |
| Primitive Roots (Historical figure: Lagrange) | p. 415 |
| The Order of an Element of Zn | p. 415 |
| Solving Polynomial Equations in Zn | p. 429 |
| Primitive Roots | p. 438 |
| Applications of Primitive Roots | p. 448 |
| Quadratic Residues (Historical figure: Eisenstein) | p. 466 |
| Squares Modulo n | p. 466 |
| Euler's Identity and the Quadratic Character of -1 | p. 478 |
| The Law of Quadratic Reciprocity | p. 489 |
| Gauss's Lemma | p. 495 |
| Quadratic Residues and Lattice Points | p. 505 |
| Proof of Quadratic Reciprocity | p. 516 |
| Primality Testing (Historical figure: Erdös) | p. 529 |
| Primality Testing | p. 529 |
| Continued Consideration of Charmichael Numbers | p. 538 |
| The Miller-Rabin Primality Test | p. 546 |
| Two Special Polynomial Equations in Zp | p. 556 |
| Proof that Miller-Rabin Is Effective | p. 561 |
| Prime Certificates | p. 573 |
| The AKS Deterministic Primality Test | p. 588 |
| Gaussian Integers (Historical figure: Euler) | p. 599 |
| Definition of the Gaussian Integers | p. 599 |
| Divisibility and Primes in Z[i] | p. 607 |
| The Division Theorem for the Gaussian Integers | p. 614 |
| Unique Factorization in Z[i] | p. 629 |
| Gaussian Primes | p. 635 |
| Fermat'sTwo Squares Theorem | p. 641 |
| Continued Fractions (Historical figure: Ramanujan) | p. 653 |
| Expressing Rational Numbers as Continued Fractions | p. 653 |
| Expressing Irrational Numbers as Continued Fractions | p. 660 |
| Approximating Irrational Numbers Using Continued Fractions | p. 673 |
| Proving That Convergents are Fantastic Approximations | p. 684 |
| Some Nonlinear Diophantine Equations (Historical figure: Germain) | p. 705 |
| Pell's Equation | p. 705 |
| Fermat's Last Theorem | p. 719 |
| Proof of Fermat's Last Theorem for n = 4 | p. 726 |
| Germain's Contributions to Fermat's Last Theorem | p. 735 |
| A Geometric Look at the Equation x4 + y4 = z2 | p. 746 |
| Index | p. 754 |
| Appendix: Axioms to Number Theory (online) | p. A-1 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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