## JSH: Pell's Equation war of words

A rather well-known in mathematical circles relation called Pell's Equation has become the focus of a bizarre to me war of words, where I've noted what I think is an interesting rational parameterization of an equation that is normally considered only with integers as it is a Diophantine equation.

The relation follows yet again for completeness of this post, though I have given it in two prior threads (would have thought one thread would be enough, but now I have to address some oddities with this one). The relation is NOT new having been known to Fermat, so it's been known for at least several hundred years:

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2)

and

x = (D + t^2)/(D - t^2)

and you get hyperbolas with D>0, the circle with D=-1, and ellipses in general with D<0.

You can see the D=-1 case from a well-known mainstream source at the following link:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17

Now one particularly belligerent poster has obsessively replied to me in prior threads, and I'm making this thread to note a few things.
1. If you do a web search on Pell's Equation—yes, I know there are things not yet on the web but a lot IS on the web so it makes sense to check—there is no mention of the rational parameterization in the first 5 links I see doing the search in Google. The 6th link though is mine!!! It's a link to my freaking math blog!!!

2. When I do the search now on Pell's Equation my math blog takes links 6 and 7 when I do that search in Google.

3. So, sorry if I'm not going to just accept it when some poster starts ranting about me being this ignorant person, as if I should just accept that it is standard teaching that Pell's Equation has this rational parameterization which can be used in the study of ellipses and hyperbolas especially when my own ideas in this area are taking over web search results!!!
Come on! Give me a break!

And I guess some of you think that's nothing. As if the world simply puts any person up to that level of attention just, oh, I don't know, just because it's such a friendly, puffy world that can just be so nice to some people, eh?

It's not a nice world. It's a mean world, with a lot of competition. The competition in this area is fierce.

If you can use Pell's Equation to classify orbits with a D number instead of eccentricity that might show something that you don't see with eccentricity. But it might not.