Combinatorial and additive number theory III CANT

Preface......Page 6
Contents......Page 7
 1 Introduction......Page 9
 2 The Group (mathbbZ/pmathbbZ)3......Page 12
 3 Proof of Theorem 3......Page 15
 References......Page 19
 1 Introduction......Page 21
 2 Modules, Lattices and Orders......Page 23
 3 The Structure of Index Forms......Page 24
 4 The Correspondence of Cubic Forms and Rings......Page 26
  5.1 The Number of Elements with Small Index in a Cubic Number Field......Page 28
  5.2 The Number of Monogenizations of a the Ring of Integers of a Cubic Number Field......Page 29
  5.3 A Positive Proportion of Cubic Orders Is Not Monogenic......Page 30
 References......Page 32
The Zeckendorf Game......Page 33
  1.2 Main Results......Page 34
  2.1 The Game Is Playable......Page 36
  2.2 Bounds on the Lengths of Games......Page 37
  2.3 Conjectures on Game Lengths......Page 39
  2.4 Winning Strategies......Page 41
 3 Future Work......Page 44
 4 Code......Page 45
 References......Page 46
 1 Introduction......Page 47
 2 Preliminary Results......Page 49
 3 Theoretical Results......Page 52
 4 Construction of 49-Iterated Riesel Numbers......Page 56
  4.2 Levels Completed Using Two Primes......Page 57
 5 Conjectures......Page 58
 References......Page 62
 1 Introduction......Page 63
  1.1 Results......Page 64
 2 Proof of Theorem2......Page 65
  3.1 Proof of Theorem1......Page 66
  3.2 Proof of Corollary1......Page 67
  3.4 Proof of Corollary3......Page 68
  3.6 Proof of Corollary5......Page 69
 References......Page 71
 1 Introduction......Page 72
 2 Preliminary Notions......Page 73
 3 Maximal Translation Invariant Filters......Page 74
 4 A Strong Version of van der Waerden's Theorem......Page 77
 5 TIFs and Left Ideals in the Space of Ultrafilters......Page 78
 References......Page 80
 1 Introduction......Page 82
 2 Coordinate Vector Matrices for Clebsch–Gordan Sums......Page 83
 3 Elementary Rules......Page 84
 4 Diagonal Zeros......Page 88
 5 Four Cases and Some Special Values......Page 93
 6 Case 1: m, n, and k Even......Page 96
 7 Case 2: m and n Even, k Odd......Page 99
 8 Parametrization of Type I and II Zeros......Page 102
 9 Cases 3 and 4: m Odd, n Even......Page 104
 References......Page 107
 1 Symmetric and Nonsymmetric Numerical Semigroups langle(n-1)k,nk,(n+1) krangle......Page 108
  1.1 Symmetric Numerical Semigroups Rn2 and Rn3......Page 110
  1.2 Nonsymmetric Numerical Semigroups Generated by Three Integers......Page 111
  1.3 Polynomial Representations for Numerical Semigroups Rnk......Page 114
 2 Numerical Semigroups Rn2, nge6......Page 115
 3 Numerical Semigroups Rn3, nge4......Page 117
  3.1 Numerical Semigroups Rn3, n=0(-5mumod5mu-2)......Page 118
  3.2 Numerical Semigroups Rn3, n=1(-5mumod5mu-2)......Page 121
 4 Extension on Numerical Semigroups Rnk, kge4......Page 124
  4.1 Conjecture and Question......Page 125
 5 Appendix: Proof of Propositions 2......Page 126
 References......Page 128
 1 Introduction......Page 129
 2 Preliminaries......Page 130
 3 Supra-SIM Sets and Their Properties......Page 132
  4.1 The Sumset Phenomenon......Page 135
  4.2 Towards B+C for SIMsets......Page 136
 References......Page 138
 1 Introduction......Page 139
 2 Main Result......Page 142
 References......Page 152
 1 Krull Dimension and Varieties......Page 153
 2 An Example of a Plane Curve......Page 155
 3 Dimension of a Monomial Hypersurface......Page 158
 4 Varieties Defined by Several Monomials......Page 161
 References......Page 166
 1 The Alternate Scaling Algorithm......Page 167
 References......Page 173
 1 Finite or Infinite?......Page 174
 2 Finitely Presented Groups......Page 175
 References......Page 179
 1 Introduction......Page 180
  2.1 Proof of Theorem2......Page 183
  2.3 A Discussion of Novelty and Bounds......Page 184
  3.1 Fourier Analysis and the Circle Method on mathbbZ......Page 185
  3.2 Density Increment Lemma......Page 187
  4.1 Proof of Lemma1......Page 189
  4.2 Proof of Lemma3......Page 190
  5.1 Major Arc Estimates......Page 192
  5.2 Proof of (24)......Page 195
  5.3 Minor Arc Estimates......Page 196
  5.4 Proof of (25)......Page 199
 References......Page 200
 1 Introduction......Page 202
  2.2 Mycielski's Theorem......Page 203
  2.4 Sierpinski's Proof......Page 204
  3.1 Kellogg's Equation......Page 205
  3.2 Equations with More Complicated Terms......Page 206
  3.3 A Theorem of Lagarias......Page 207
 4 Appendix: Nonstandard Extensions of mathbbR......Page 209
 References......Page 212
 1 Introduction......Page 213
  2.1 Notation......Page 214
  2.2 Argument Overview......Page 215
 3 Circuits with the 3x+1 Dynamical System......Page 216
  3.2 Weighted Binomial Coefficients......Page 218
  3.3 Dual-Radix Modular Division......Page 225
  3.4 Circuits in the 3x-1 Dynamical System......Page 226
 References......Page 229
 1 Introduction......Page 231
 3 Degree 2......Page 233
 4 Degree 1......Page 235
 References......Page 237