数学天书中的证明 原书第4版 英文PDF电子书下载

NumberTheory 1

1．Sixproofs of the infinity of primes 3

2．Bertrand'spostulate 7

3．Binomial coefficients are(almost)never powers 13

4．Representing numbers as sums of two squares 17

5．The law of quadratic reciprocity 23

6．Every finite division ringis a field 31

7．Some irrational numbers 35

8．Three times π2/6 43

Geometry 51

9．Hilbert's third problem：decomposing polyhedra 53

10．Lines in the plane and decompositions of graphs 63

11．The slope problem 69

12．Three applications of Euler's formula 75

13．Cauchy's rigidity theorem 81

14．Touching simplices 85

15．Every large point set has an obtuse angle 89

16．Borsuk's conjecture 95

Analysis 101

17．Sets,functions,and the continuum hypothesis 103

18．Inpraise of inequalities 119

19．The fundamental theorem of algebra 127

20．One square and an odd number of triangles 131

21．Atheoremof Pólya on polynomials 139

22．On a lemma of Littlewood and Offord 145

23．Cotangent and the Herglotz trick 149

24．Buffon's needle problem 155

Combinatorics 159

25．Pigeon-hole and double counting 161

26．Tiling rectangles 173

27．Three famous theorems on finite sets 179

28．Shuffling cards 185

29．Lattice paths and determinants 195

30．Cayley's formula for the number of trees 201

31．Identities versus bijections 207

32．Completing Latin squares 213

Graph Theory 219

33．The Dinitz problem 221

34．Five-coloring plane graphs 227

35．How to guard a museum 231

36．Turán's graph theorem 235

37．Communicating without errors 241

38．The chromatic number of Kneser graphs 251

39．Of friends and politicians 257

40．Probability makes counting(sometimes)easy 261

About the Illustrations 270

Index 271