Essays on Numbers and Figures

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American Mathematical Soc., 2000 - Mathematics - 75 pages
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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.
 

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Contents

Conjugate Numbers
1
Rational Parametrizations of the Circle
5
Sums of Squares of Polynomials
7
Representing Numbers as the Sum of Two Squares
9
Can Any Knot Be Unraveled?
11
Construction of a Regular 17gon
15
The Markov Equation
19
IntegerValued Polynomials
21
Intersection Points of the Diagonals of Regular Polygons
33
The Chromatic Polynomial of a Graph
39
Brocard Points
43
Diophantine Equations for Polynomials
51
The Pascal Lines
55
One Butterfly and Two Butterflies Theorems
59
The Van der Waerden Theorem on Arithmetical Progressions
61
Isogonal Conjugate Points
65

Chebyshev Polynomials
23
Vectors in Geometry
25
The Averaging Method and Geometric Inequalities
27
Cubic Curves Related to the Triangle
71
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