**Mathematical World**

Volume: 16;
2000;
75 pp;
Softcover

MSC: Primary 00;

Print ISBN: 978-0-8218-1944-9

Product Code: MAWRLD/16

List Price: $19.00

AMS Member Price: $15.20

MAA Member Price: $17.10

**Electronic ISBN: 978-1-4704-2479-4
Product Code: MAWRLD/16.E**

List Price: $19.00

AMS Member Price: $15.20

MAA Member Price: $17.10

# Essays on Numbers and Figures

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*V. V. Prasolov*

This is the English translation of the book originally
published in Russian. It contains 20 essays, each dealing with a
separate mathematical topic. The topics range from brilliant
mathematical statements with interesting proofs, to simple and
effective methods of problem-solving, to interesting properties of
polynomials, to exceptional points of the triangle. Many of the topics
have a long and interesting history. The author has lectured on them
to students worldwide.

The essays are independent of one another for the most part, and
each presents a vivid mathematical result that led to current research
in number theory, geometry, polynomial algebra, or
topology.

#### Readership

Advanced high school and undergraduate students, mathematics educators at secondary and university level; general mathematical audience.

#### Reviews & Endorsements

This volume is quite delightful … Each essay … engages the reader's interest in an immediate and lively manner … can be highly recommended for well-prepared students aspiring to a top-level mathematics degree and will prove a marvelous source for teachers … a good bedtime read for Putnam competitors … a pleasure indeed for the general lover of mathematics.

-- Mathematical Reviews

A worthwhile resource for mathematics departments to have on hand … this collection serves as a nice resource for accessible topics beyond the standard curriculum around the level of calculus.

-- MAA Online

#### Table of Contents

# Table of Contents

## Essays on Numbers and Figures

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Foreword ix10 free
- Chapter 1. Conjugate Numbers 112 free
- Chapter 2. Rational Parametrizations of the Circle 516
- Chapter 3. Sums of Squares of Polynomials 718
- Chapter 4. Representing Numbers as the Sum of Two Squares 920
- Chapter 5. Can Any Knot Be Unraveled? 1122
- Chapter 6. Construction of a Regular 17-gon 1526
- Chapter 7. The Markov Equation 1930
- Chapter 8. Integer-Valued Polynomials 2132
- Chapter 9. Chebyshev Polynomials 2334
- Chapter 10. Vectors in Geometry 2536
- Chapter 11. The Averaging Method and Geometric Inequalities 2738
- Chapter 12. Intersection Points of the Diagonals of Regular Polygons 3344
- Chapter 13. The Chromatic Polynomial of a Graph 3950
- Chapter 14. Brocard Points 4354
- Chapter 15. Diophantine Equations for Polynomials 5162
- Chapter 16. The Pascal Lines 5566
- Chapter 17. One Butterfly and Two Butterflies Theorems 5970
- Chapter 18. The Van der Waerden Theorem on Arithmetical Progression 6172
- Chapter 19. Isogonal Conjugate Points 6576
- Chapter 20. Cubic Curves Related to the Triangle 7182
- Back Cover Back Cover187