Straight from the Book

Title page......Page 1
Foreword......Page 4
Preface......Page 6
Contents......Page 12
1 Some Useful Substitutions......Page 15
1.1 The relation a2 + b2 + c2 = abc + 4......Page 16
1.2 The relations abc = a + b + c + 2 and ab + bc + ca + 2abc = 1......Page 23
1.3 The relation a2 + 62 + c2 + 2abc = 1......Page 35
1.4 Notes......Page 41
2 Always Cauchy-Schwarz.........Page 43
2.1 Notes......Page 75
Addendum 2.A Cauchy-Schwarz in Number Theory......Page 76
3 Look at the Exponent......Page 105
3.1 Introduction......Page 105
3.2 Local-global principle......Page 106
3.3 Legendre's formula......Page 110
3.4 Problems with combinatorial and valuation-theoretic aspects......Page 118
3.5 Lifting exponent lemma......Page 124
3.6 p-adic techniques......Page 130
3.7 Miscellaneous problems......Page 139
3.8 Notes......Page 148
Addendum 3.A Classical Estimates on Prime Numbers......Page 149
Addendum 3.B An Introduction to p-adic Numbers......Page 155
4 Primes and Squares......Page 203
4.1 Notes......Page 217
5 T2's Lemma......Page 219
5.1 Notes......Page 240
Addendum 5A Holder's Inequality in Action......Page 241
6 Some Classical Problems in Extremal Graph Theory......Page 249
6.1 Notes......Page 265
Addendum 6.A Some Pearls of Extremal Graph Theory......Page 266
Addendum 6.B Probabilities in Combinatorics......Page 279
7 Complex Combinatorics......Page 295
7.1 Tiling and coloring problems......Page 295
7.2 Counting problems......Page 299
7.3 Miscellaneous problems......Page 313
7.4 Notes......Page 323
Addendum 7.A Finite Fourier Analysis......Page 324
8 Formal Series Revisited......Page 351
8.1 Counting problems......Page 353
8.2 Proving identities using generating functions......Page 363
8.3 Recurrence relations......Page 368
8.4 Additive properties......Page 375
8.5 Miscellaneous problems......Page 385
8.6 Notes......Page 389
Addendum 8.A Lagrange's Inversion Theorem......Page 391
9 A Little Introduction to Algebraic Number Theory......Page 409
9.1 Tools from linear algebra......Page 410
9.2 Cyclotomy......Page 416
9.3 The gcd trick......Page 422
9.4 The theorem of symmetric polynomials......Page 424
9.5 Ideal theory and local methods......Page 434
9.6 Miscellaneous problems......Page 440
9.7 Notes......Page 447
Addendum 9.A Equations over Finite Fields......Page 448
Addendum 9.B A Glimpse of Algebraic Number Theory......Page 470
10 Arithmetic Properties of Polynomials......Page 499
10.1 The a - b\f (a)-f(b) trick......Page 499
10.2 Derivatives and p-adic Taylor expansions......Page 508
10.3 Hilbert polynomials and Mahler expansions......Page 511
10.4 p-adic estimates......Page 520
10.5 Miscellaneous problems......Page 527
10.6 Notes......Page 534
11 Lagrange Interpolation Formula......Page 535
11.1 Notes......Page 551
12 Higher Algebra in Combinatorics......Page 553
12.1 The determinant trick......Page 555
12.2 Matrices over F2......Page 560
12.3 Applications of bilinear algebra......Page 566
12.4 Matrix equations......Page 575
12.5 The linear independence trick......Page 582
12.6 Applications to geometry......Page 590
12.7 Notes......Page 597
Bibliography......Page 599