From: "John R Ramsden" <jr@redmink.demon.co.uk>
Subject: Re: representing integers by quadratic forms
Date: Sat, 20 Mar 1999 19:30:13 -0800
Newsgroups: sci.math

Alan Williams-Key wrote in message <7cp0n5$5n6$1@news8.svr.pol.co.uk>...
>
> Can anyone tell me where I can find out the theories about the
> representation of numbers by quadratic forms a*x^2+bxy+c*y^2?
>
> Thanks
>
> Alan


Here are some refs, all of which I heartily recommend:

1. "Binary Quadratic Forms. Classical Theory and Modern Computations",
    D A Buell, Springer-Verlag, 1989, ISBN 0-387-97037-1 or 3-540-97037-1.
   (don't know what the difference is, hardback v. paperback?)

    (Very good for computation purposes, as the title suggests.)

2. "The Sensual Quadratic Form", John Horton Conway, Carus Mathematical
   Monographs, number 26, 1997, ISBN 0-88385-030-3.

  (This is the book in which J H Conway makes a good case for defining
  -1 as prime! He also defines a "topograph", which is a diagrammatic way
  of reading off many key properties of quadratic forms)

3. "Quadratic Forms", G N Watson, (quoting off the top of my head).

   This was the last complete coverage of spinor genera by elementary
   (and complicated) techniques. More recent books use the abstract
   methods developed around the time this book appeared.

   Actually, G N Watson, although a distinguished mathematician, was
   noted for his "busy work" ability. Among other achievements he was
   the first to solve the Square Pyramid problem, by using an incredibly
   intricate descent argument involving elliptic functions. This problem,
   solved only a few years ago by elementary methods, asks for all
   integer solutions to:

      1^2 + 2^2 + ... + m^2  =  n^2

   He was also an arch-train-spotter, and knew all the UK railway
   timetables off by heart (no mean achievement before the 1960s!)
   In fact he was railroaded by the Government into suggesting
   improvements to the schedules.

4. Number Theory books published by R D Carmichael and L E Dickson
    (Dover Books.)


Cheers

John R Ramsden  (jr@redmink.demon.co.uk)