数学天书中的证明 : 第4版

Number Theory

1.Six proofs of the infinity of primes

2.Bertrand's postulate

3.Binomialcoefficients are (almost) never powers

4.Representing numbers as sums of two squares

5.The law of quadratic reciprocity

6.Every firute division ring is afield

7.Some irrational numbers

8.Three times ∏2/6

Geometry

9.Hilbert's tlurd problem: decomposing polyhedra

10.Lines in the plane and decompositions of graphs

11.The slopeproblem

12.Three applications of Euler's formula

13.Cauchy's rigidity theorem

14.Touching simplices

15.Every large point set has an obtuse angle

16.Borsuk's conjecture

Analysis

17.In praise ofinequalities

19.The fundamental theorem of algebra

20.One square and an odd number of triangles

21.A theorem of Polya on polynomials

22.On alemma of Littlewood and Offord

23.Cotangent and the Herglotz trick

24.Buffon's needle problem

Combinatorics

25.Pigeon-hole and double counting

26.Tiling rectangles

27.Three famous theorems on finite sets

28.Shuffling cards

29.Lattice paths and determinants

30.Cayley's formula for the number of trees

31.IdenMes versus bijections

32.Completing Latin squares

Graph Theory

33.The Dinitz problem

34.Five-coloring plane graphs

35.How to guard a museum

36.Turan's graph theorem

37.Communicating without errors

38.The chromatic number of Kneser graphs

39.Of friends and politicians

40.Probability makes counting (sometimes) easy

About the Illustrations

Index

Paul Erdos liked to talk about The Book，in which God maintains the perfect proofs for mathematical theorems， following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdos also said that you need not believe in God but， as a mathematician， you should believe in The Book. A few years ago， we suggested to lum to write up a first （and very modest） approximation to The Book. He was enthusiastic about the idea and， characteristically， went to work immediately， filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdos' 85th birthday. With Paul's unfortunate death in the summer of 1996， he is not listed as a co-author. Instead this book is dedicated to his memory.