Ebook: Number Theory: New York Seminar 1991–1995

This volume is dedicated to Harvey Cohn, Distinguished Professor Emeritus of Mathematics at City College (CUNY). Harvey was one of the organizers of the New York Number Theory Seminar, and was deeply involved in all aspects of the Seminar from its first meeting in January, 1982, until his retirement in December, 1995. We wish him good health and continued hapiness and success in mathematics. The papers in this volume are revised and expanded versions of lectures delivered in the New York Number Theory Seminar. The Seminar meets weekly at the Graduate School and University Center of the City University of New York (CUNY). In addition, some of the papers in this book were presented at a conference on Combinatorial Number Theory that the New York Number Theory Seminar organized at Lehman College (CUNY). Here is a short description of the papers in this volume. The paper of R. T. Bumby focuses on "elementary" fast algorithms in sums of two and four squares. The actual talk had been accompanied by dazzling computer demonstrations. The detailed review of H. Cohn describes the construction of modular equations as the basis of studies of modular forms in the one-dimensional and Hilbert cases.

---

Contents: Sums of Four Squares.- On the number of co-prime-free sets.- The primary role of modular equations.- Approximation Methods in Transcendental Function Computations and Some Physical Applications.- Diophantine Approximation Problem arising from VLSI Design.- Linear Diophantine Problems.- The smallest maximal set of pairwise disjoint partitions.- Sum set Cardinalities of Line Restricted Planar Sets.- On solvability of a system of two Boolean Linear Equations.- Brauer number and twisted Fermat Motives.- A Remark on a Paper of Erdos and Nathanson.- Towards a classification of Hilbert Modular Threefolds.- Special Theta Relations.- Minimal bases and g-adic Representations of Integers.- Finite Graphs and the Number of Sums and Products.- Hilbert's theorem 94 and function fields.- Some Applications of Probability to additive Number Theory and Harmonic Analysis.- Quadratic Irrationals and Continued Fractions.- Progression Bases for Finite Cyclic Groups.- Sums of Finite Sets.- Four Squares with Few Squares.

---

Contents: Sums of Four Squares.- On the number of co-prime-free sets.- The primary role of modular equations.- Approximation Methods in Transcendental Function Computations and Some Physical Applications.- Diophantine Approximation Problem arising from VLSI Design.- Linear Diophantine Problems.- The smallest maximal set of pairwise disjoint partitions.- Sum set Cardinalities of Line Restricted Planar Sets.- On solvability of a system of two Boolean Linear Equations.- Brauer number and twisted Fermat Motives.- A Remark on a Paper of Erdos and Nathanson.- Towards a classification of Hilbert Modular Threefolds.- Special Theta Relations.- Minimal bases and g-adic Representations of Integers.- Finite Graphs and the Number of Sums and Products.- Hilbert's theorem 94 and function fields.- Some Applications of Probability to additive Number Theory and Harmonic Analysis.- Quadratic Irrationals and Continued Fractions.- Progression Bases for Finite Cyclic Groups.- Sums of Finite Sets.- Four Squares with Few Squares.
Content:
Front Matter....Pages i-x
Sums of Four Squares....Pages 1-8
On the Number of Co-Prime-Free Sets....Pages 9-18
The Primary Role of Modular Equations....Pages 19-41
Approximation Methods in Transcendental Function Computations and Some Physical Applications....Pages 43-69
Diophantine Approximation Problem Arising from VLSI Design....Pages 71-90
Linear Diophantine Problems....Pages 91-95
On the Sum of the Reciprocals of the Differences Between Consecutive Primes....Pages 97-101
The Smallest Maximal Set of Pairwise Disjoint Partitions....Pages 103-113
Sum Set Cardinalities of Line Restricted Planar Sets....Pages 115-133
On Solvability of a System of Two Boolean Linear Equations....Pages 135-150
“Brauer Numbers” of Twisted Fermat Motives....Pages 151-175
A Remark on a Paper of Erd?s and Nathanson....Pages 177-179
Towards a Classification of Hilbert Modular Threefolds....Pages 181-193
Special Theta Relations....Pages 195-199
Minimal Bases and g-Adic Representations of Integers....Pages 201-209
Finite Graphs and the Number of Sums and Products....Pages 211-219
Hilberth’s Theorem 94 and Function Fields....Pages 221-228
Some Applications of Probability to Additive Number Theory and Harmonic Analysis....Pages 229-251
Quadratic Irrationals and Continued Fractions....Pages 253-268
Progression Bases for Finite Cyclic Groups....Pages 269-279
Sums of Finite Sets....Pages 281-293
Four Squares with Few Squares....Pages 295-297

---

Contents: Sums of Four Squares.- On the number of co-prime-free sets.- The primary role of modular equations.- Approximation Methods in Transcendental Function Computations and Some Physical Applications.- Diophantine Approximation Problem arising from VLSI Design.- Linear Diophantine Problems.- The smallest maximal set of pairwise disjoint partitions.- Sum set Cardinalities of Line Restricted Planar Sets.- On solvability of a system of two Boolean Linear Equations.- Brauer number and twisted Fermat Motives.- A Remark on a Paper of Erdos and Nathanson.- Towards a classification of Hilbert Modular Threefolds.- Special Theta Relations.- Minimal bases and g-adic Representations of Integers.- Finite Graphs and the Number of Sums and Products.- Hilbert's theorem 94 and function fields.- Some Applications of Probability to additive Number Theory and Harmonic Analysis.- Quadratic Irrationals and Continued Fractions.- Progression Bases for Finite Cyclic Groups.- Sums of Finite Sets.- Four Squares with Few Squares.
Content:
Front Matter....Pages i-x
Sums of Four Squares....Pages 1-8
On the Number of Co-Prime-Free Sets....Pages 9-18
The Primary Role of Modular Equations....Pages 19-41
Approximation Methods in Transcendental Function Computations and Some Physical Applications....Pages 43-69
Diophantine Approximation Problem Arising from VLSI Design....Pages 71-90
Linear Diophantine Problems....Pages 91-95
On the Sum of the Reciprocals of the Differences Between Consecutive Primes....Pages 97-101
The Smallest Maximal Set of Pairwise Disjoint Partitions....Pages 103-113
Sum Set Cardinalities of Line Restricted Planar Sets....Pages 115-133
On Solvability of a System of Two Boolean Linear Equations....Pages 135-150
“Brauer Numbers” of Twisted Fermat Motives....Pages 151-175
A Remark on a Paper of Erd?s and Nathanson....Pages 177-179
Towards a Classification of Hilbert Modular Threefolds....Pages 181-193
Special Theta Relations....Pages 195-199
Minimal Bases and g-Adic Representations of Integers....Pages 201-209
Finite Graphs and the Number of Sums and Products....Pages 211-219
Hilberth’s Theorem 94 and Function Fields....Pages 221-228
Some Applications of Probability to Additive Number Theory and Harmonic Analysis....Pages 229-251
Quadratic Irrationals and Continued Fractions....Pages 253-268
Progression Bases for Finite Cyclic Groups....Pages 269-279
Sums of Finite Sets....Pages 281-293
Four Squares with Few Squares....Pages 295-297
....