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Topic: Parametric Pell's Equation and circle
Replies: 17   Last Post: Apr 9, 2009 2:11 AM

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 quasi Posts: 9,080 Registered: 7/15/05
Re: Parametric Pell's Equation and circle
Posted: Apr 6, 2009 10:53 PM

On Mon, 6 Apr 2009 19:39:44 -0700 (PDT), marcus_bruckner@yahoo.com
wrote:

>On Apr 3, 11:30 pm, JSH <jst...@gmail.com> wrote:
>> On Apr 3, 9:44 pm, quasi <qu...@null.set> wrote:
>>

>> > On Fri, 3 Apr 2009 19:40:15 -0700 (PDT), JSH <jst...@gmail.com> wrote:
>> > >So, the world has its first known rational parameterization of
>> > >ellipses. And, oh yeah, of course, hyperbolas with the regular D>1.

>>
>> > >It's amazing how close their parametric solutions are to the circle
>>
>> > >How was this result not known for thousands of years?
>>
>> > Rational parameterizations for ellipses and hyperbolas have been known
>> > for hundreds of years. I think it was Euler who developed the basic

>>
>> Well yes. That part of what I said was wrong.
>>
>> However, this particular rational parameterization is new.
>>

>> > method for finding such parameterizations.
>>
>> > Just do a Google search.
>>
>>
>> > <http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/curves/rational.html>
>>
>> > Here's another:
>>
>> > <http://en.wikipedia.org/wiki/Algebraic_curve>
>>
>> > quasi
>>
>>
>> And I did reach claiming no rational parameterizations had been done
>> before for ellipses and hyperbolas.
>>
>> There never has been a *single* parameterization that handles circles,
>> ellipses, and hyperbolas just by sign before:
>>
>> With x^2 - Dy^2 = 1
>>
>> the parameterization:
>>
>> y = 2[f_2*v - 1]/[D - (f_2*v - 1)^2]
>>
>> and
>>
>> x = [D + (f_2*v - 1)^2]/[D - (f_2*v - 1)^2]
>>
>> where f_2 is an integer factor of D-1, you get a parameterization for
>> hyperbolas with integer D>1, for circles with D=-1, and for ellipses
>> with D<-1.
>>
>> It is a unifying result. And a world's first.
>>
>> James Harris

>
> Move over little dog. I have just discovered the most
>freaking nifty result ever seen on the planet. I have discovered
>that lots of numbers can be expressed as sums of squares in more
>than one way.
>
> Here is my beautiful awe-inspiring example: N = 850.
>
> Note that 850 = 3^2 + 29^2
> = 121^2 + 27^2
> = 15^2 + 25^2.
>
> Yep, that's right. Count 'em. THREE freaking different
>ways! Even more astounding, 845 can ALSO be written as a sum
>of two squares in three different ways! And so can 725!
>
> No one else on earth has ever discovered this. I know
>because I did a Google search on the phrase "My discovery
>on sums of squares". Nothing! PROOF that nobody has
>ever thought of this before!
>
> IT'S HUGE!!!!
> It's NIFTY!!!!
> It's STUNNING!!!!
>
> I am the king. Lying jealous welfare queen mathematicians
>will try to deny this. I am a MAJOR DISCOVERER. I am a problem
>solver.
>
> I must rush off now and add this to my blog. Oh wait. I don't
>have a blog. I must rush off and create a blog. Plus I must
>write this up for submission to the Annals.
>
> And THINK, people. THINK how many avenues for questions this
>opens up. Let s(N) be the number of ways that N can be
>written as a sum of two squares. What is the limit of
>s(N) / log(N) as N --> infinity? NO ONE KNOWS! Given M, what
>is the expected value of the smallest N such that s(N) = M ?
>NO ONE KNOWS! Think of all the questions! This is BRAINSTORMING
>MATH, people! Cutting edge! Nifty! Freaking amazing! Beautiful!
>
> Ow! I just got a terrible bruise in right between my shoulder
>blades where I was patting myself on the back too hard! Oh,
>the perils of Genius! But you freaking little lying LOSERS
>will never know what THAT feels like! A whole new area of
>research just opened up and you worthless inert fools just
>sit there! Only a Genius like JSH is going to appreciate
>this. He might even have a way to connect it to Pell's
>Equation! Wow!

Haha -- that's a good one!

quasi

Date Subject Author
4/3/09 jstevh@gmail.com
4/3/09 waste of time
4/3/09 jstevh@gmail.com
4/3/09 quasi
4/4/09 rossum
4/4/09 jstevh@gmail.com
4/6/09 marcus_b
4/6/09 quasi
4/7/09 jstevh@gmail.com
4/8/09 Patrick Karl
4/9/09 jstevh@gmail.com
4/4/09 David Bernier
4/4/09 jstevh@gmail.com
4/4/09 jstevh@gmail.com
4/4/09 willo_thewisp@hotmail.com
4/4/09 jstevh@gmail.com
4/5/09 rossum
4/5/09 jstevh@gmail.com