Proofs from THE BOOK

B́a trước
Springer Science & Business Media, 29 thg 6, 2013 - 199 trang
The (mathematical) heroes of this book are "perfect proofs": brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory.
Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erdös, who himself suggested many of the topics in this collection.
The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background.
 

Nội dung

Number Theory
2
Six proofs of the infinity of primes
3
Bertrands postulate
7
Binomial coefficients are almost never powers
13
Representing numbers as sums of two squares
17
Every finite division ring is a field
23
Some irrational numbers
27
Geometry
35
In praise of inequalities
101
A theorem of Pólya on polynomials
109
On a lemma of Littlewood and Offord
117
Combinatorics
121
Three famous theorems on finite sets
135
Cayleys formula for the number of trees 141
140
Completing Latin squares
147
The Dinitz problem
153

Lines in the plane and decompositions of graphs
45
The slope problem
51
Three applications of Eulers formula
57
Cauchys rigidity theorem
63
The problem of the thirteen spheres
67
Touching simplices
73
Every large point set has an obtuse angle
77
Borsuks conjecture
83
Analysis
89
Sets functions and the continuum hypothesis 91
90
Graph Theory
159
Fivecoloring plane graphs
161
How to guard a museum
165
Turáns graph theorem
169
Communicating without errors
173
Of friends and politicians
183
Probability makes counting sometimes easy 187
186
About the Illustrations
196
Index
197
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