Number Theory : A Lively Introduction with

Number Theory: A Mathemythical Approach, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Many theorems are preceded by Numerical Proof Previews, which are numerical examples that will help give students a concrete understanding of both the statements of the theorems and the ideas behind their proofs, before the statement and proof are formalized in more abstract terms. In addition, many applications of number theory are explained in detail throughout the text, including some that have rarely (if ever) appeared in textbooks. A unique feature of the book is that every chapter includes a math myth, a fictional story that introduces an important number theory topic in a friendly, inviting manner. Many of the exercise sets include in-depth Explorations, in which a series of exercises develop a topic that is related to the material in the section.

James Pommersheim is a Professor of Mathematics at Reed College in Portland, Oregon. In addition to teaching at Reed since 2004, he has served as an instructor for the Johns Hopkins University Center for Talented Youth. Dr. Pommersheim also serves on the Reed faculty committees for Research on Human Subjects and Academic Support Services.

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 x⁴ + y⁴ = z²	p. 746
Index	p. 754
Appendix: Axioms to Number Theory (online)	p. A-1
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Number Theory : A Lively Introduction with Proofs, Applications, and Stories

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