Number Theory : A Lively Introduction with Proofs, Applications, and Stories

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  • Format: Hardcover
  • Copyright: 2010-02-01
  • Publisher: Wiley

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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.

Author Biography

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.

Table of Contents

Prefacep. viii
Structure of the Textp. ix
To the Studentp. x
To the Instructorp. xi
Acknowledgementsp. xiv
Prologue: Number Theory Through the Agesp. xvi
Numbers, Rational and Irrational (Historical figures: Pythagoras and Hypatia)p. 6
Numbers and the Greeksp. 6
Numbers You Knowp. 13
A First Look at Proofsp. 17
Irrationality of ?2p. 28
Using Quantifiersp. 32
Mathematical Induction (Historical figure: Noether)p. 42
The Principle of Mathematical Inductionp. 42
Strong Induction and the Well-Ordering Principlep. 55
The Fibonacci Sequence and the Golden Ratiop. 67
The Legend of the Golden Ratiop. 76
Divisibility and Primes (Historical figure: Eratosthenes)p. 92
Basic Properties of Divisibilityp. 92
Prime and Composite Numbersp. 98
Patterns in the Primesp. 104
Common Divisors and Common Multiplesp. 116
The Division Theoremp. 124
Applications of god and 1cmp. 138
The Euclidean Algorithm (Historical figure: Euclid)p. 148
The Euclidean Algorithmp. 148
Finding the Greatest Common Divisorp. 156
A Greeker Argument that ?2 Is Irrationalp. 172
Linear Diophantine Equations (Historical figure: Diophantus)p. 182
The Equation aX + bY= 1p. 182
Using the Euclidean Algorithm to Find a Solutionp. 191
The Diophantine Equation aX + bY = np. 200
Finding All Solutions to a Linear Diophantine Equationp. 205
The Fundamental Theorem of Arithmetic (Historical figure: Mersenne)p. 216
The Fundamental Theoremp. 216
Consequences of the Fundamental Theoremp. 225
Modular Arithmetic (Historical figure: Gauss)p. 241
Congruence Modulo np. 241
Arithmetic with Congruencesp. 254
Check-Digit Schemesp. 267
The Chinese Remainder Theoremp. 274
The Gregorian Calendarp. 288
The Mayan Calendarp. 296
Modular Number Systems (Historical figure: Turing)p. 307
The Number System Zn: An Informal Viewp. 307
The Number System Zn: Definition and Basic Propertiesp. 310
Multiplicative Inverses in Znp. 322
Elementary Cryptographyp. 338
Encryption Using Modular Multiplicationp. 343
Exponents Modulo n (Historical figure: Fermat)p. 355
Fermat's Little Theoremp. 355
Reduced Residues and the Euler ?-Functionp. 368
Euler's Theoremp. 379
Exponentiation Ciphers with a Prime Modulusp. 390
The RSA Encryption Algorithmp. 399
Primitive Roots (Historical figure: Lagrange)p. 415
The Order of an Element of Znp. 415
Solving Polynomial Equations in Znp. 429
Primitive Rootsp. 438
Applications of Primitive Rootsp. 448
Quadratic Residues (Historical figure: Eisenstein)p. 466
Squares Modulo np. 466
Euler's Identity and the Quadratic Character of -1p. 478
The Law of Quadratic Reciprocityp. 489
Gauss's Lemmap. 495
Quadratic Residues and Lattice Pointsp. 505
Proof of Quadratic Reciprocityp. 516
Primality Testing (Historical figure: Erdös)p. 529
Primality Testingp. 529
Continued Consideration of Charmichael Numbersp. 538
The Miller-Rabin Primality Testp. 546
Two Special Polynomial Equations in Zpp. 556
Proof that Miller-Rabin Is Effectivep. 561
Prime Certificatesp. 573
The AKS Deterministic Primality Testp. 588
Gaussian Integers (Historical figure: Euler)p. 599
Definition of the Gaussian Integersp. 599
Divisibility and Primes in Z[i]p. 607
The Division Theorem for the Gaussian Integersp. 614
Unique Factorization in Z[i]p. 629
Gaussian Primesp. 635
Fermat'sTwo Squares Theoremp. 641
Continued Fractions (Historical figure: Ramanujan)p. 653
Expressing Rational Numbers as Continued Fractionsp. 653
Expressing Irrational Numbers as Continued Fractionsp. 660
Approximating Irrational Numbers Using Continued Fractionsp. 673
Proving That Convergents are Fantastic Approximationsp. 684
Some Nonlinear Diophantine Equations (Historical figure: Germain)p. 705
Pell's Equationp. 705
Fermat's Last Theoremp. 719
Proof of Fermat's Last Theorem for n = 4p. 726
Germain's Contributions to Fermat's Last Theoremp. 735
A Geometric Look at the Equation x4 + y4 = z2p. 746
Indexp. 754
Appendix: Axioms to Number Theory (online)p. A-1
Table of Contents provided by Ingram. All Rights Reserved.

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