Sunday, March 29, 2009


JSH: Pell's Equation alternates

So remarkably in a well-worked area where supposedly no simple new research results were available I've found that for every odd prime D, there are three alternates to Pell's Equation, which also solve it, for which only one is true for a particular D:

j^2 - Dk^2 = -1

where x = 2Dk^2 - 1 = 2j^2 + 1


j^2 - Dk^2 = -2

where x = Dk^2 - 1 = j^2 + 1


j^2 - Dk^2 = 2

where x = Dk^2 + 1 = j^2 - 1

and x^2 - Dy^2 = 1.

What's remarkable then is that past mathematicians were approaching Pell's Equation BACKWARDS when D is a prime, as the solutions above for j are approximately the square root of x, so they're easier to find!!!

Here's an example, with D=19:

13^2 - 19*(3)^2 = -2

So j=13. With that alternate I have:

x = Dk^2 - 1 = j^2 + 1, so x = 13^2 + 1 = 170

And 170^2 - 19*(39)^2 = 1, as required.

Of interest now then is checking the mathematical literature to see what has been said about these alternates and it appears that only the so-called negative Pell's Equation has gotten attention with research done attempting to find its frequency.

I suggest readers search on: negative Pell's Equation

But an immediate guess from the result shown here is that x^2 - Dy^2 = -1, is available for an odd prime D approximately 1/3 of the time.

So that is an immediate research route to check the distribution for an ODD PRIME D, as I'd guess that researchers were simply looking at D an integer, including composites.

An astounding mystery now though is, why is this result new?

That is, however, not a question for mathematical research but one for mathematical scholars.

Readers looking to find out more about these equations and how they are derived should go to my math blog:

I highly recommend doing searches or going to the library and checking on Pell's Equation first.

I think it's a remarkable result but unfortunately, for me, it's just another major research find, where in the past I've faced resistance from the mathematical community.

I hope that is not the case here, as hey, math is fun! And in this area you can compare your own research findings to that of past mathematicians over 2000 years of human history.

Now when is the last time you could so something like that?

The door is open. There is no way of knowing at this point what major result may lurk around the corner from paths taken with this line of research. But the math community has to be willing to step through…

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