Re: JSH: Binary quadratic Diophantines

On Sep 20, 7:40 pm, MichaelW <ms...@xxxxxxxxxx> wrote:
Thankyou to all me defenders and detractors. First thing James is
right about, let's get back to the maths. I don't want to put too much

Well I've read your entire post, and don't want to just wander off, so
I'll reply to it,
as I think you show sincere interest in it, so I won't just leave you
hanging.

But I think my original post in this thread covered the bases rather
well, while
you've gone off on a bit of a tangent, but that's ok, as it's Usenet.

And I'm intrigued by explaining a few more things.

information into one post so what I am going to do is to go through an
example and try and bring what I am saying and what James is saying a
little closer together. Some terminology to start; Step 1 algorithm
refers to the transformation of a quadratic equation in two variables
into a simpler form. Step 2 algorithm solves the simpler form. This
post is purely about Step 1. Also apologies ahead of time for any
mistakes. It has been a long time since I delved into matrix algebra
so the rust may well show!

My sampler is

x^2+xy+y^2=1+x+y  (1)

Not standard notation but never mind. Solutions for (x,y) are (1,1),
(-1,1) and (1,-1). The algorithm translates that into

t^2 + 3s^2 = 64  (2)

where t=(3-2(x+y)). Now (1) is an ellipse centred at (1/3,1/3) and
rotated 45 degress clockwise. We can apply the normal transformation
matrices to obtain

p^2 + 3q^2 = 8/3  (3)

I didn't check your work. But I'll note that what I give is easier
than doing a rotation using matrices.

It's also kind of wild to me that you got that 8/3. With Diophantine
equations that's a no-no.

No fractions.

For reference seehttp://en.wikipedia.org/wiki/Rotation_(mathematics)#In_two_dimensions.
Obviously (2) and (3) are not entirely the same but nor are they
entirely different. The second thing James is right about is when he
says
<quote>
Integers add some quirks, which I'm sorry I have to say goes to the
question of real expertise in this area, as I'm seeing replies from
posters like you and "MichaelW" which indicate you don't understand
that Diophantine equations ARE special in various ways.
</quote>
Right about the fact we are solving for integers which does add some
complications. I have been puzzling over this during the weekend; it

Good! It's fascinating mathematics. Diophantine equations have
puzzled people
for thousands of years.

They are good mental exercise to consider, especially since you can
play with
them more easily in some ways because you're using integers, but they
can have
incredible complexities to them that defy standard analytical methods.

has been made somewhat harder in that I don't know exactly how the
values in the Step 1 algorithm for A, B and C were derived. Never

I derived them using tautological spaces. It's explained on my math
blog.

Those curious can just Google: binary quadratic Diophantine equations

The link that comes up #1 when I do that search (and presumably will
for you as well) is to my
blog post which also has a link to the derivation.

(I don't post direct links to my blogs any more as it's easier to just
use Google searches!!!)

I get bored for a while with what follows so will jump to where it
gets interesting again.

mind, onward and upward!

Let us revisit equation (2). We have t=(2-3x-3y) and as it turns out s=
(x-y). This can be expressed as a matrix transformation (sorry about
the formatting):

[t] = [-3 -3] [x] + [2]   (4)
[s]   [1  -1] [y]    [0]

Now the matrix we use to multiply (x,y) does not look like a standard
rotation matrix. For typing convenince set a=sqrt(2)/2. The rotation
matrix that I am using is

[cos(45) sin(45)] =  [a  a] = lets call it R.
[-sin(45) cos(45)]    [-a a]

Now we have

[-3 -3] = -a * R X [2 1]    (5)
[1  -1]                [1 2]

So there is a relationship the Step1 algorithm transformation and the
more traditional approach which keeps the shape and proportions of the
original curve.

This is a far as I have got so far; basically I am working backwards
from the results so there is a fair bit of reverse engineering going
on here. I think I have undermined myself by using an example equation
that is symmetrical around x=y. For example in (5) the -a multiplier
at the front may actually be -1/2a. As time permits I will look into a
non-symmetrical equation.

One last point for James. The mathematical basis for this post is
generally considered somewhat basic and textbooks may or may not refer
to it since we are operating here in two dimensions. Physicists (and
game designers!) work in 3D so most of the heavy lifting in both
transformation and solving quadratic equations is done with at least
three variables. If you want to get more pick up from the scientists
you should generalise your work to (x,y,z).

The algebra doesn't care. It is probably rather easy to move to 3
variables and in fact I
was doing 3 variables to start and used z=1, to get the general
approach to binary quadratic
Diophantine equations as I noticed it was in easy reach.

Of course, with 3 variables it's not necessarily true that as simple
an answer is given as with 2.

But the world will not know until someone does the math. And the
world has not motivated me
to do it, as having your work repeatedly ignored and trashed is not a
great motivator.

So yes, it is possible it could be done. Certainly tautological
spaces do not care, and can easily
handle 3 variables just like they did 2, when I was working with close
to the 3 variable solution anyway.

It seems weird to me that my world cares so little about great
mathematical advances, but I do not
make the world. I live in it.

I cannot force people to care about great mathematics.


James Harris

.

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