Saturday, August 29, 2009


JSH: Categorization conflict with Pell's Equation?

I've been amazed by finding out that a rather simple rational parameterization of the equation known as Pell's Equation has been known for hundreds of years, when I found it by re-discovering it, and I'm bringing this subject up again—as I have another thread on it—as I'm wondering if it is a case where because mathematicians traditionally look at Pell's Equation as a Diophantine equation, the rational parameterization of it has been dismissed and almost forgotten though it can be found if you search for it and possibly keeps getting re-discovered:

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

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


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:
See: eqns. 16 & 17

If I'd seen that as a kid in standard texts when I liked to do things like have the computer graph curves I'd have definitely programmed that one, as it lets you go from hyperbolas to ellipses by simply changing a single number.

It would be amazing if mathematicians by categorizing Pell's Equation as a Diophantine equation created an odd situation where the human species kind of just happens to keep forgetting that you have this way to look at it with rationals!!! Over the entire planet. The entire human species.

You see, the rational parameterization is, of course, useless for solving Pell's Equation in integers only, which is what you do considering it as a Diophantine equation, and in fact, Fermat dismissed it for that reason in one of his letters, hundreds of years ago, when he and others were engaging in contests around what we now call Pell's Equation.

They'd compete to get solutions in integers for various D's that are hard.

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