From: jr@redmink.demon.co.uk (John R Ramsden)
Subject: Re: quadratic diophantine equation
Date: Sun, 01 Aug 1999 18:05:55 GMT
Newsgroups: sci.math
Keywords: representation by quadratic forms

On 30 Jul 1999 20:03:22 GMT, heye@netcologne.de (Thomas Heye) wrote:

>I'm interested in the numbers generated by
>$a^2 +-ab +b^2$, where a, b are integers. I transformed it to
>a^2 -ab +b^2 =(a +b/2)^2 +3(a -b/2)^2
>or
>a^2 +ab +b^2 =(a -b/2)^2 +3(a +b/2)^2
>This reminds me of an ellipse equation ... Maybe someone can help me
>or give me some hints.
>
>Regards,
>
>Thomas

If (a, b) = 1, i.e. a, b are relatively prime then a^2 +a.b + b^2
is a product of primes of the form 3.Z + 1. This can easily be seen
by multiplying by four, completing the square, and noting that the
square of an integer is never of the form 3.Z + 2.

The converse also holds, i.e. any integer whose prime factors are
all of the form 3.Z + 1 can be represented as a^2 + a.b + b^2 for
integers a, b (in a number of ways that depends on the number of
factors, prime or otherwise, of the number).

For two integers p, q of the form a^2 + a.b + b^2, c^2 + c.d + d^2
their product is of the form e^2 + e.f + f^2 where e, f are defined
as follows in which w^2 + w + 1 = 0:

   e - w.f  =  (a - w.b).(c - w.d)

i.e. equating rational and irrational parts:

   e, f  =  a.c - b.d,  a.d + b.c + b.d

For example:

   1^2 + 1.1 + 1^2 = 3

   2^2 + 2.1 + 1^2 = 7

So using:

   (1 - w).(2 - w)  =  2 - 3.w + w^2

                    =  2 - 3.w - (w + 1)

                    =  1 - 4.w

we find, sure enough:

   1^2 + 1.4 + 4^2  =  21  =  3.7

The only way you can get even numbers of the form a^2 + a.b + b.^2,
or numbers of that form divisible by primes of the form 3.Z + 2,
is for a, b to have common factors (2 or the primes 3.Z + 2 resp).

A good modern reference, with plenty of numerical examples is
"Quadratic Forms", by Beuller [sp?], Springer-Verlag.


Cheers

---

John R Ramsden             #   "No one who has not shared a submarine
                           #    with a camel can claim to have plumbed
 (jr@redmink.demon.co.uk)  #    the depths of human misery."
                           #
                           #      Ritter von Haske
                           #        "Adventures of a U-boat Commander".
[duplicate sig deleted -- djr