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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 Z_{n}: An Informal View | p. 307 |
The Number System Z_{n}: Definition and Basic Properties | p. 310 |
Multiplicative Inverses in Z_{n} | 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 Z_{n} | p. 415 |
Solving Polynomial Equations in Z_{n} | 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 Z_{p} | 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 x^{4} + y^{4} = z^{2} | p. 746 |
Index | p. 754 |
Appendix: Axioms to Number Theory (online) | p. A-1 |
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