Monday, July 18, 2011

Two conics equation

A remarkable little equation has a fascinating story:

x2 - Dy2 = 1

In the field of rationals it can be solved as a parametric equation:

x = (D + t2)/(D - t2), y = 2t/(D - t2)

And notice that it gives two of the conic sections, as with D less than 0 it gives ellipses, giving the circle itself with D=-1.

With D greater than 0 it gives hyperbolas.

I have no idea how useful it is for, say, drawing those conic sections and haven't seen much on the subject on the web. But it can be used to draw, and D should be related to eccentricity.

Looking now at integer only solutions, so it is a Diophantine equation, which just means integers only, I'll introduce a simple relation.

Given x2 - Dy2 = F, it must be true that:

(x+Dy)2 - D(x+y)2 = -F(D-1)

You can verify the correctness of the second result by multiplying out:

x2 + 2Dxy +D2y2 - D(x2 + 2xy + y2) = -FD + F


x2 + 2Dxy +D2y2 - Dx2 - 2Dxy - Dy2 = -FD + F

so 2Dxy cancels out, and I can move things around to group like so:

x2 - Dy2 - Dx2 + D2y2 + FD = F

and notice I have my original equation negatively mirrored inside which is easier to see like so:

x2 - Dy2 - D(x2 - Dy2 - F) = F, so: x2 - Dy2 = F

So we've verified the relation as valid.

And I can use that relation over and over again to get a series. So now let's use it with our two conics equation, which means F=1, and let's use D=2, so now I can get a series, where here are the first five:

1. x2 - 2y2 = 1

2. (x+2y)2 - 2(x+y)2 = -1

3. (3x+4y)2 - 2(2x + 3y)2 = 1

4. (7x + 10y)2 - 2(5x + 7y)2 = -1

5. (17x + 24y)2 - 2(12x + 17y)2 = 1

so you can use the simple case of x=1 and y = 0:

1. 12 = 1

2. (1)2 - 2(1)2 = -1

3. (3)2 - 2(2)2 = 1

4. (7)2 - 2(5)2 = -1

5. (17)2 - 2(12)2 = 1

and get solutions.

That trick, however, only works to give you answers so easily for D=2.

For the curious I've delved far deeper into its use with the two conics equation, and found I could connect the two conics equation to what are called Pythagorean Triples.

And I used it to help explain why some integer solutions to the two conics equation are much bigger or smaller than others. A clue is the number of prime factors of D-1. If D-1 is prime then solutions tend to be small, but if it has a lot of small prime factors, and especially if 4 is a factor then solutions get much larger.

(Editing to note that subsequent research gave the full picture, while at this point I still just had one key part of it.)

For example a famous case is D=61, where 60 makes it big:

17663190492 - 61(226153980)2 = 1

But notice that for D=62, because 61 is prime it is much smaller:

632 - 62(8)2 = 1

It is possible to trivially prove that an infinite number of integer solutions for x and y exist, using one more equation.


(n2 + 1)2 - (n2 + 2)n2 = 1

multiplying out gives:

n4 + 2n2 + 1 - n4 - 2n2 = 1

So it simplifies to 1=1, so it's called an identity. In mathematics an equation is called an identity if it simplifies to a simple equality, like x=x or y=y, or 1=1. Notice then we can just let:

x = n2 + 1, y = n, and have a solution if D = n2 + 2

For example, if n=3, D = 11, and 102 - 11(3)2 = 1.

There are other such identities for the curious, and you can guess at them from the one shown.

What I call here the two conics equation is generally considered by mathematicians with integers only as a Diophantine equation, where "Diophantine" just means, integers only, so then D is usually an integer greater than 1, and mathematicians traditionally call the equation Pell's Equation.

But they also note that the name is a mis-attribution to a guy named John Pell. That is, he shouldn't have the equation named after him based on any contributions which he did, so it was a mistake for that to happen, and usually Euler is blamed for that mistake!

So the puzzle is, why don't mathematicians just quit calling it Pell's Equation if that is a mistake?

For years I've followed their tradition, but their tradition also includes ignoring the rational parametric solution, as they just focus on Diophantine equations, which may be a historical oddity as well.

Turns out that Fermat played with finding integer solutions as a competitive game with others and they thought the parametric equation was too easy for that game, as it made finding answers trivial. Finding integer answers only is harder. Their focus on integers only as a game became locked into a mathematical preference by the math community.

Since that was in the 17th century and near four hundred years ago, it seems to me it makes sense in the 21st century to just let that oddity go, use a more explanatory name, and at least mention the parametric solution. Remarkably mathematicians tend to refuse to even mention the parametric equation, except for the circle case. See: eqns. 16 & 17

Other disciplines are capable of letting go past mistakes.

James Harris
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