proofs

Recommendations

Info

Bertrand’s postulate 9 times. Here each summand is at most 1, since it satisfies ⌊ ⌋ ⌊ ⌋ 2n n p k − 2 p k < 2n ( ) n p k − 2 p k − 1 = 2, and it is an integer. Furthermore the summands vanish whenever p k > 2n. Thus ( ) 2n n contains p exactly ∑ (⌊ ⌋ ⌊ ⌋) 2n n p k − 2 p k ≤ max{r : p r ≤ 2n} k≥1 times. Hence the largest power of p that divides ( ) 2n n is not larger than 2n. In particular, primes p > √ 2n appear at most once in ( ) 2n n . Furthermore — and this, according to Erdős, is the key fact for his proof — primes p that satisfy 2 3 n < p ≤ n do not divide ( ) Examples such as 2n n at all! Indeed, `26 3p > 2n implies (for n ≥ 3, and hence p ≥ 3) that p and 2p are the only multiples of p that appear as factors in the numerator of (2n)! n!n! , while we get two p-factors in the denominator. (4) Now we are ready to estimate ( ) 2n n . For n ≥ 3, using an estimate from page 12 for the lower bound, we get 4 n 2n ≤ ( 2n n ) ≤ ∏ p≤ √ 2n 2n · ∏ √ 2n
10 Bertrand’s postulate One can extract even more from this type of estimates: From (2) one can derive with the same methods that ∏ p ≥ 2 1 30 n for n ≥ 4000, n 1 1 dt = log n t
Page 2: Martin Aigner Günter M. Ziegler Pr
Page 5 and 6: Prof. Dr. Martin Aigner FB Mathemat
Page 7 and 8: VI Preface to the Fourth Edition Wh
Page 9 and 10: VIII Table of Contents 21. A theore
Page 12 and 13: Six proofs of the infinity of prime
Page 14 and 15: Six proofs of the infinity of prime
Page 16 and 17: Bertrand’s postulate Chapter 2 We
Page 20 and 21: Bertrand’s postulate 11 by compar
Page 22 and 23: Binomial coefficients are (almost)
Page 24 and 25: Binomial coefficients are (almost)
Page 26 and 27: Representing numbers as sums of two
Page 32 and 33: The law of quadratic reciprocity Ch
Page 34 and 35: The law of quadratic reciprocity 25
Page 40 and 41: Every finite division ring is a fie
Page 42 and 43: Every finite division ring is a fie
Page 44 and 45: Some irrational numbers Chapter 7
Page 46 and 47: Some irrational numbers 37 and this
Page 48 and 49: Some irrational numbers 39 F(x) may
Page 50: Some irrational numbers 41 Now assu
Page 53 and 54: 44 Three times π 2 /6 v 1 1 2 y v
Page 55 and 56: 46 Three times π 2 /6 For m = 1, 2
Page 57 and 58: 48 Three times π 2 /6 1 f(t) = 1 t
Page 59 and 60: 50 Three times π 2 /6 (3) It has b
Page 62 and 63: Hilbert’s third problem: decompos
Page 68 and 69:
Hilbert’s third problem: decompos
Page 70:
Hilbert’s third problem: decompos
Page 73 and 74:
64 Lines in the plane, and decompos
Page 75 and 76:
66 Lines in the plane, and decompos
Page 78 and 79:
The slope problem Chapter 11 Try fo
Page 80 and 81:
The slope problem 71 • Every move
Page 82:
The slope problem 73 For the second
Page 85 and 86:
76 Three applications of Euler’s
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
82 Cauchy’s rigidity theorem Q: Q
Page 93 and 94:
84 Cauchy’s rigidity theorem A be
Page 95 and 96:
86 Touching simplices l l l ′ Pr
Page 97 and 98:
88 Touching simplices For our examp
Page 99 and 100:
90 Every large point set has an obt
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
96 Borsuk’s conjecture Theorem. L
Page 107 and 108:
98 Borsuk’s conjecture On the oth
Page 109 and 110:
100 Borsuk’s conjecture To obtain
Page 112 and 113:
Sets, functions, and the continuum
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
In praise of inequalities Chapter 1
Page 130 and 131:
In praise of inequalities 121 where
Page 132 and 133:
In praise of inequalities 123 Proo
Page 134:
In praise of inequalities 125 Seco
Page 137 and 138:
128 The fundamental theorem of alge
Page 140 and 141:
One square and an odd number of tri
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
A theorem of Pólya on polynomials
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
On a lemma of Littlewood and Offord
Page 156 and 157:
On a lemma of Littlewood and Offord
Page 158 and 159:
Cotangent and the Herglotz trick Ch
Page 160 and 161:
Cotangent and the Herglotz trick 15
Page 162 and 163:
Cotangent and the Herglotz trick 15
Page 164 and 165:
Buffon’s needle problem Chapter 2
Page 166 and 167:
Buffon’s needle problem 157 The c
Page 168:
Combinatorics 25 Pigeon-hole and do
Page 171 and 172:
162 Pigeon-hole and double counting
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 182 and 183:
Tiling rectangles Chapter 26 Some m
Page 184 and 185:
Tiling rectangles 175 Third proof.
Page 186:
Tiling rectangles 177 References [1
Page 189 and 190:
180 Three famous theorems on finite
Page 191 and 192:
182 Three famous theorems on finite
Page 194 and 195:
Shuffling cards Chapter 28 How ofte
Page 196 and 197:
Shuffling cards 187 Indeed, if A i
Page 198 and 199:
Shuffling cards 189 Let T be the nu
Page 200 and 201:
Shuffling cards 191 Theorem 1. Let
Page 202 and 203:
Shuffling cards 193 Proof. (1) We
Page 204 and 205:
Lattice paths and determinants Chap
Page 206 and 207:
Lattice paths and determinants 197
Page 208 and 209:
Lattice paths and determinants 199
Page 210 and 211:
Cayley’s formula for the number o
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Identities versus bijections Chapte
Page 218 and 219:
Identities versus bijections 209 Pr
Page 220 and 221:
Identities versus bijections 211 As
Page 222 and 223:
Completing Latin squares Chapter 32
Page 224 and 225:
Completing Latin squares 215 Proof
Page 226 and 227:
Completing Latin squares 217 be dis
Page 228:
Graph Theory 33 The Dinitz problem
Page 231 and 232:
222 The Dinitz problem In 1976 Vizi
Page 233 and 234:
224 The Dinitz problem X A bipartit
Page 235 and 236:
226 The Dinitz problem G : L(G) : a
Page 237 and 238:
228 Five-coloring plane graphs From
Page 239 and 240:
230 Five-coloring plane graphs is s
Page 241 and 242:
232 How to guard a museum The follo
Page 243 and 244:
234 How to guard a museum “Museum
Page 245 and 246:
236 Turán’s graph theorem Let us
Page 247 and 248:
238 Turán’s graph theorem Thus b
Page 250 and 251:
Communicating without errors Chapte
Page 252 and 253:
Communicating without errors 243 cl
Page 254 and 255:
Communicating without errors 245 Le
Page 256 and 257:
Communicating without errors 247 On
Page 258 and 259:
Communicating without errors 249 No
Page 260 and 261:
The chromatic number of Kneser grap
Page 262 and 263:
Page 264:
Page 267 and 268:
258 Of friends and politicians w 1
Page 270 and 271:
Probability makes counting (sometim
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Proofs from THE BOOK 269
Page 280 and 281:
Index acyclic directed graph, 196 a
Page 282 and 283:
Index 273 matrix-tree theorem, 203
show all

proofs

Create successful ePaper yourself

Delete template?

Save as template?