Saturday, May 02, 2009


JSH: Pell's equation and question of math fraud

I've been pondering the situation where I found that what is commonly called Pell's Equation, has this parametric rational solution that math people don't talk about, and also has this connection to these alternate equations that make solving them easy, or vice versa, you can solve Pell's Equation with them, and now I can get some resolution to troubling questions.

Like, I have various math discoveries, which other people say are not math discoveries, and I've considered that hey, maybe I'm evolutionarily more advanced, so I can understand things that less advanced people can't, which is more simply captured by noting that dogs can't learn calculus. Their brains make that impossible. If I were evolutionarily advanced then I could have math discoveries simple to me that others could no more understand than a dog could understand calculus.

That was a scary scenario, and I'm happy to put it to rest, which is what Pell's Equation has allowed.

Pell's Equation is just x^2 - Dy^2 = 1, where math people are looking for integers that fit into all those boxes, so D is a positive integer, and x and y are integers. 9^2 - 5*4^2 = 1, is an example of such an outcome.

The equation has been known for thousands of years or something.

Well in our greatly advanced times, it seems mainstream math people have not bothered to note that if D is a prime number such that D = 1 mod 4, then another equation often called the negative Pell's Equation is solved by Pell's Equation:

j^2 - Dk^2 = -1, j = sqrt((x-1)/2)

So with my example above with D=5, notice that j = sqrt((9-1)/2) = 2, works, as 2^2 - 5*1^2 = -1.

And you could go the OTHER way, and find the solution to Pell's Equation, as j = sqrt((x-1)/2) means that

x = 2j^2 + 1, and x = 2*2^2 + 1 = 9, as required.

That always works. It's actually a rather trivial result mathematically which you can figure out from the main Pell's Equation as

x^2 - Dy^2 = 1, means that x^2 - 1 = Dy^2, so (x-1)(x+1) = Dy^2,

and if D is prime then it can only be a factor of x-1 or x+1, meaning that one of them must be either a square or 2 times a square as if x is odd then they both are even. You can work out all the alternates by noting how D can divide across. Trivial algebra.

That's not hard to understand, but if you go on the web and look at math texts on Pell's Equation you will not find that result:'s_equation


You may think there is a trick in D being prime, where D = 1 mod 4, but turns out there are two more equations like the negative Pell's Equation if D does not equal 1 mod 4, and one more on top of those if D is a composite in a special way.

So there are 4 other alternates to Pell's Equation, where any of those will solve Pell's Equation if they exist, and the solution is smaller.

Here's a fun example to show the size difference:

1766319049^2 - 61*226153980^2 = 1

but, consider

j^2 - Dk^2 = -1

as you get a solution to x, when x^2 - Dy^2 = 1, with x = 2j^2 + 1, and with D=61, notice an astounding difference in the size of the solution—

29718^2 - 61*3805^2 = -1

and a quick check with your computer's calculator will show that, yes:

1766319049 = 2*297182 + 1.

You can use continued fractions to solve in either direction which is neat because the way math people teach continued fractions you may naively think there is only one answer they give, but math people traditionally go ONE WAY, using all positives.

The reality of the simpler solution for the negative Pell's Equation indicates they're going the wrong way, so using continued fractions properly—using negatives—could greatly simplify solving Pell's Equation by working to give one of the alternates sooner, and then allowing one to pull its answer from them.

Ok. That's not rocket science. It's relatively simple algebra that I assume you can all understand. I'm not trying to teach calculus to dogs here. I'm not some super advanced mutant freak who can understand things you cannot, which means, fraud is now on the table, as I've talked about the above for a while now.

The Wikipedia has not been updated. Wolfram hasn't updated. Far as I know math people are still teaching Pell's Equation like they always have, and emails on this subject to people like Arjen Lenstra, have not been answered. He's an expert in this area.

Near as I can tell math people have no intentions of ever acknowledging the above. Ever.

Now then, are they dogs who can't learn calculus, or are they academics who are playing a dangerous game, certain they can win, and damn humanity in the process?

You try and figure it out. It's your world as well as mine. If there is fraud on this level, you can be very certain it's very bad, and the reasons are terrifying and not at all good for the future of the entire human race.

After all, they're being rather daring at this point.

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