Ebook: Representations of Integers as Sums of Squares
Author: Emil Grosswald (auth.)
- Tags: Number Theory
- Year: 1985
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
- pdf
During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but con sisted of particularly gifted and eager listeners; unfortunately, their back ground varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the inten tion of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-4
Preliminaries....Pages 5-12
Sums of Two Squares....Pages 13-23
Triangular Numbers and the Representation of Integers as Sums of Four Squares....Pages 24-37
Representations as Sums of Three Squares....Pages 38-65
Legendre’s Theorem....Pages 66-71
Representations of Integers as Sums of Nonvanishing Squares....Pages 72-83
The Problem of the Uniqueness of Essentially Distinct Representations....Pages 84-90
Theta Functions....Pages 91-106
Representations of Integers as Sums of an Even Number of Squares....Pages 107-127
Various Results on Representations as Sums of Squares....Pages 128-133
Preliminaries to the Circle Method and the Method of Modular Functions....Pages 134-148
The Circle Method....Pages 149-174
Alternative Methods for Evaluating r s (n)....Pages 175-187
Recent Work....Pages 188-218
Back Matter....Pages 219-251
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-4
Preliminaries....Pages 5-12
Sums of Two Squares....Pages 13-23
Triangular Numbers and the Representation of Integers as Sums of Four Squares....Pages 24-37
Representations as Sums of Three Squares....Pages 38-65
Legendre’s Theorem....Pages 66-71
Representations of Integers as Sums of Nonvanishing Squares....Pages 72-83
The Problem of the Uniqueness of Essentially Distinct Representations....Pages 84-90
Theta Functions....Pages 91-106
Representations of Integers as Sums of an Even Number of Squares....Pages 107-127
Various Results on Representations as Sums of Squares....Pages 128-133
Preliminaries to the Circle Method and the Method of Modular Functions....Pages 134-148
The Circle Method....Pages 149-174
Alternative Methods for Evaluating r s (n)....Pages 175-187
Recent Work....Pages 188-218
Back Matter....Pages 219-251
....