## JSH: Advancing theory of continued fractions

There is a reason for the discussions I've raised about Pell's Equation and alternates like the negative Pell's Equation besides demonstrating a woeful disdain for mathematical knowledge that some of you have, as the real prize is advancing the methodology of continued fractions. And that is a HUGE prize. Unlike any other available, as then we can see further on the shoulders of the giants before us, like Gauss and Lagrange.

You see, for the mathematically astute of you, there is a problem with Pell's Equation solutions leading to solutions for the alternates as the alternates ALL have solutions that are smaller than their corresponding Pell's Equation solution by roughly a square.

For instance, for j^2 - Dk^2 = -1, any solution MUST give a solution to x^2 - Dy^2 = 1, with x = 2j^2 + 1.

So you can see the square with j^2, and now we get to continued fractions!!!

Because you can solve Pell's Equation using continued fractions. But you can solve the negative Pell's Equation—when it exists—using continued fractions as well!!!

But it's SMALLER by a square!

I've hypothesized that people normally do continued fractions in the least efficient way, using all positives, when you should use negatives.

If so, then you can find a convergent much faster, meaning that you can advance the use of continued fractions by a huge margin.

I've been puzzling over this area for a while and thought recently, hey, why not toss it out there.

There are some dark forces among you though, so I played it out for a few days for you to understand who they are, and what they are.

They are, quite simply, anti-mathematicians.

They pretend to be mathematicians, but show their true colors when discovery is about, as their role is to block human progress!!!

They were put on this earth to make a challenge of it! For the great ones.

They were put on this earth to trip YOU up, as only the truly great among you can get past them.