Re: JSH: Increasingly the world's math reference

On Sep 11, 12:09 am, MichaelW <ms...@xxxxxxxxxx> wrote:
On Thu, 10 Sep 2009 22:27:46 -0700, JSH wrote:
On Sep 10, 1:02 am, MichaelW <ms...@xxxxxxxxxx> wrote:
Where can I get this solution? I checked the blog but nothing seemed to
exactly match.

I find myself muddling through those posts myself.  When I wrote that
post last year it all seemed so clear.

I find portions of my math blog are nearly incomprehensible to me now.

Just puzzling over what I call surrogate factoring took me days to
finally get it all back into focus.  And by days I mean like a week and
a half.

Freaking thing has information packed so tightly that maybe it takes a
genius to understand it easily, or I don't think there is a human being
on the planet who can just read through much of my blog.

I've sat down to look over a single page and after hours, given up.

Thanks, Michael W.

My suggestion to you: walk away.  If the mathematics doesn't grab you
then it's not for you anyway.

The solution to binary quadratic Diophantine equations is a result of
the use of what I call tautological spaces.

Google: tautological spaces

They can be very hard to grasp.  Weirdly so as it's just algebra.  But
make no mistake, you may not be able to fully comprehend what is in
front of you, which can be odd, but you yourself note you cannot see the
solution.

But a world is seeing it, which is what I mean by pushing people to
objective checks.

You will not understand everything that is presented to you in this
world.  That's nothing of which to be ashamed.

Save your energy, if you can't understand it, then that's ok.

James Harris

Actually what I said was that there was no general solution posted that I
could find. It turns out the Google links to the wrong page, at least for
me. Following the advice in the original post of this thread I googled
"solving binary quadratic Diophantine equations" and got the October 18,
2008 entry which is close but suggests a solution only when B=0.

Nope. That's the correct page. World selected it.

What you're seeing is a world selection, not mine, as reflected by
what comes up highly in Google.

World is kind of smart, you know?

It picked the correct page.

I noticed a link at the beginning to here:

http://docs.google.com/Doc?id=dfs4mntm_35gc7sc4dv

Part three of that document links to the September 26, 2008 article which
seems to be what you are referring to.

Hey, the world can pick ANY page it wants. And yes, I'm fascinated by
what the world picks as it's kind of weird to see the entire world in
that way and weirder paying attention to what it picks!

Thanks to web search engines though you can see what the world picks
out of one particular person's research as there are 187 posts on my
math blog.

The world is being VERY selective about which of that number it wants,
which are the few that come up highly in certain Google searches.

For instance I promoted my prime counting research a LOT more than
this latest research, and would even post links to it in the past, but
do a search on: prime counting function

I don't come up in the top 10.

But do a search on: binary quadratic Diophantine equations

And I have #1.

Note that this article starts off with an equation in three variables
(x,y,z) but immediately sets z to 1 for a more traditional quadratic
equation.

To get a binary quadratic.

It is actually fairly straightforward to follow (just algebra as you say)
and I have run a few equations.

Yup, easy algebra, but still not necessarily so easy to comprehend as
you repeatedly are noting difficulties, and got really confused on the
page the world picked as you thought it only solved B=0. It does not.

For example an elliptical equation:

x^2 + x*y + y^2 = x + y + 1 (equation 1)

using the algorithm generates

t^2 + 3 * s^2 = 16          (equation 2)

where t = (3x+3y-2) and s is "some integer". This is of course correct
since the actual solutions ( (x,y) = (1,1), (-1,1) or (-1,1)) all
generate an integer solution for s (s = 0, 2 and 2 respectively). However
this is still an elliptical equation. Cyclically applying the algorithm
still generates elliptical equations only with larger terms. There does
not appear to be an approach to resolving the elliptical form in your
algorithm.

There are two parts, part 1 shows how every binary quadratic
Diophantine equation is connected to a simpler one of the form:

u^2 + Dv^2 = F

The second part talks about how to solve for u and v.

The Dario page's methods for resolving elliptical quadratic Diophantine
equations suggests simplifying and then scanning the full range of
possible integer values. For example in equation 2 above since s^2 equals
(16 - t^2)/3 then s as an integer can only be in the range (-2..2). This
is fine for small values but for really large co-efficients is hardly
efficient.

No doubt I have made a mistake somewhere. Could you indicate how you
would resolve a Diophantine equation of the form

Ax^2 + By^2 = C

where A, B, C are positive integers. For the wider maths community what
is the "official" (for lack of a better word) best algorithm for such an
equation?

Regards, Michael W.

It's on the page. It's actually a lift!

I don't know why you think that page just works for B=0.

You can use the Chinese Remainder Theorem but I went with what I call
my little congruence result.

It's right there on the page! I rely on a simple identity:

Given: x^2 + Dy^2 = F,

(x-Dy)^2 + D(x+y)^2 = F(D+1)

which is why it's a lift, as you repeat using that identity you get a
higher power of D+1, for instance, next is:

((1-D)x-2Dy)^2 + D(2x + (1-D)y)^2 = F(D+1)^2

so you can solve for u and v, with u^2 + Dv^2 = F(D+1)^j, where j is
arbitrarily high using some simple congruence mathematics (I hope),
which I give on the page, as I'm simply explaining the concept here.

That is actually easily checkable by anyone now just from this
mathematics as it's just using that identity, which is easily verified
by multiplying out.

VERY EASY algebra. And I want to remind the page was world selected
as revealed by Google.

And physics people are about objective tests, so I want you to have
the objective test that is not about wishing or hoping or endless
discussion, as you can just do a Google search!!!

Google: solving binary quadratic Diophantine equations

Or just: binary quadratic Diophantine equations

The world is picking the page for a reason.


James Harris

.

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