Binary Quadratic Forms

Classical Theory and Modern Computations

  • Duncan A. Buell

Table of contents

  1. Front Matter
    Pages i-x
  2. Duncan A. Buell
    Pages 1-12
  3. Duncan A. Buell
    Pages 13-20
  4. Duncan A. Buell
    Pages 21-48
  5. Duncan A. Buell
    Pages 49-75
  6. Duncan A. Buell
    Pages 77-85
  7. Duncan A. Buell
    Pages 87-107
  8. Duncan A. Buell
    Pages 109-133
  9. Duncan A. Buell
    Pages 135-157
  10. Duncan A. Buell
    Pages 159-189
  11. Duncan A. Buell
    Pages 191-212
  12. Back Matter
    Pages 213-247

About this book

Introduction

The first coherent exposition of the theory of binary quadratic forms was given by Gauss in the Disqnisitiones Arithmeticae. During the nine­ teenth century, as the theory of ideals and the rudiments of algebraic number theory were developed, it became clear that this theory of bi­ nary quadratic forms, so elementary and computationally explicit, was indeed just a special case of a much more elega,nt and abstract theory which, unfortunately, is not computationally explicit. In recent years the original theory has been laid aside. Gauss's proofs, which involved brute force computations that can be done in what is essentially a two­ dimensional vector space, have been dropped in favor of n-dimensional arguments which prove the general theorems of algebraic number the­ ory. In consequence, this elegant, yet pleasantly simple, theory has been neglected even as some of its results have become extremely useful in certain computations. I find this neglect unfortunate, because binary quadratic forms have two distinct attractions. First, the subject involves explicit computa­ tion and many of the computer programs can be quite simple. The use of computers in experimenting with examples is both meaningful and enjoyable; one can actually discover interesting results by com­ puting examples, noticing patterns in the "data," and then proving that the patterns result from the conclusion of some provable theorem.

Keywords

Arithmetic Finite algebra calculus equation number theory proof theorem

Authors and affiliations

  • Duncan A. Buell
    • 1
  1. 1.Supercomputing Research CenterBowieUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-4542-1
  • Copyright Information Springer-Verlag New York 1989
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-8870-1
  • Online ISBN 978-1-4612-4542-1
  • About this book