Combinatorial Number Theory: Proceedings of the "Integers Conference 2011", Carrollton, Georgia, USA, October 26-29, 2011

by Bruce Landman (Editor), Melvyn B. Nathanson (Editor), Jaroslav Nesetril (Editor), Richard J. Nowakowski (Editor), Carl Pomerance (Editor)

Overview

This volume contains selected refereed papers based on lectures presented at the "Integers Conference 2011", an international conference in combinatorial number theory that was held in Carrollton, Georgia, United States in October 2011. This was the fifth Integers Conference, held bi-annually since 2003. It featured plenary lectures presented by Ken Ono, Carla Savage, Laszlo Szekely, Frank Thorne, and Julia Wolf, along with sixty other research talks.

This volume consists of ten refereed articles, which are expanded and revised versions of talks presented at the conference. They represent a broad range of topics in the areas of number theory and combinatorics including multiplicative number theory, additive number theory, game theory, Ramsey theory, enumerative combinatorics, elementary number theory, the theory of partitions, and integer sequences.

Product Details

ISBN-13:	9783110280487
Publisher:	De Gruyter
Publication date:	08/19/2013
Series:	De Gruyter Proceedings in Mathematics
Pages:	166
Product dimensions:	6.69(w) x 9.45(h) x 0.02(d)
Age Range:	18 Years

About the Author

Bruce M. Landman, University of West Georgia, Carrollton, USA; Melvyn B. Nathanson, The City University of New York, Bronx, USA; Jaroslav Nešetril, Charles University, Prague, Czech Republic; Richard J. Nowakowski, Dalhousie University, Halifax, Canada; Carl Pomerance, Dartmouth College, Hanover, ; Aaron Robertson, Colgate University, Hamilton, USA.

Table of Contents

Preface v

1 The Misère Monoid of One-Handed Alternating Games Rebecca Milley Richard J. Nowakowski Paul Ottaway 1

1.1 Introduction 1

1.1.1 Background 2

1.2 Equivalences 4

1.3 Outcomes 10

1.4 The Misère Monoid 12

2 Images of C-Sets and Related Large Sets under Nonhomogeneous Spectra Neil Hindman John H. Johnson 15

2.1 Introduction 15

2.2 The Various Notions of Size 19

2.3 The Functions f_α and h_α 25

2.4 Preservation of J-Sets, C-Sets, and C*-Sets 27

2.5 Preservation of Ideals 33

3 On the Differences Between Consecutive Prime Numbers, I Daniel A. Goldston Andrew H. Ledoan 37

3.1 Introduction and Statement of Results 37

3.2 The Hardy-Littlewood Prime k-Tuple Conjectures 37

3.3 Inclusion-Exclusion for Consecutive Prime Numbers 39

3.4 Proof of the Theorem 42

4 On Sets of Integers Which Are Both Sum-Free and Product-Free Pär Kurlberg Jeffrey C. Lagarias Carl Pomerance 45

4.1 Introduction 45

4.2 The Upper Density 47

4.3 An Upper Bound for the Density in Z/nZ 50

4.4 Examples With Large Density 51

5 Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems Frank Thome 55

5.1 Introduction 55

5.2 Counting Fields in General 56

5.2.1 Counting Torsion Elements in Class Groups 59

5.3 Davenport-Heilbronn, Delone-Faddeev, and the Main Terms 60

5.3.1 The Work of Belabas, Bhargava, and Pomerance 61

5.4 The Four Approaches 62

5.5 The Shintani Zeta-Function Approach 63

5.5.1 Nonequidistribution in Arithmetic Progressions 66

5.6 A Refined Geometric Approach 67

5.6.1 Origin of the Secondary Term 68

5.6.2 A Correspondence for Cubic Forms 69

5.7 Equidistribution of Heegner Points 70

5.7.1 Heegner Points and Equidistribution 71

5.8 Hirzebruch Surfaces and the Maroni Invariant 73

5.9 Conclusion 74

6 Spotted Tilings and n-Color Compositions Brian Hopkins 79

6.1 Background 79

6.2 n-Color Composition Enumerations 81

6.3 Conjugable H-Color Compositions 86

7 A Class of Wythoff-Like Games Aviezri S. Fraenkel Yuval Tanny 91

7.1 Introduction 91

7.2 Constant Function 93

7.2.1 A Numeration System 94

7.2.2 Strategy Tractability and Structure of the P-Positions 98

7.3 Superadditive Functions 99

7.4 Polynomial 103

7.5 Further Work 106

8 On the Multiplicative Order of F_n+1/F_n Modulo F_m Takao Komatsu Florian Luca Yohei Tachiya 109

8.1 Introduction 109

8.2 Preliminary Results 110

8.3 Proof of Theorem 8.1 114

8.4 Comments and Numerical Results 120

9 Outcomes of Partizan Euclid Neil A. McKay Richard J. Nowakowski 123

9.1 Introduction 123

9.2 Game Tree Structure 125

9.3 Reducing the Signature 128

9.3.1 Algorithm 132

9.4 Outcome Observations 133

9.5 Open Questions 134

10 Lecture Hall Partitions and the Wreath Products C_k $$ S_n Thomas W. Pensyl Carla D. Savage 137

10.1 Introduction 137

10.2 Lecture Hall Partitions 138

10.3 Statistics on C_k $$ S_n 139

10.4 Statistics on s-Inversion Sequences 140

10.5 From Statistics on C_k $$ S_n to Statistics on I_n,k 141

10.6 Lecture Hall Polytopes and s-Inversion Sequences 143

10.7 Lecture Hall Partitions and the Inversion Sequences I_n,k 145

10.8 A Lecture Hall Statistic on C_k $$ S_n 148

10.9 Inflated Eulerian Polynomials for C_k $$ S_n 150

10.10 Concluding Remarks 153

Index 155