Starting from the basic reduced binary quadratic form:

u

^{2}+ Dv

^{2}= F

the simple result is:

(u-Dv)

^{2}+ D(u+v)

^{2}= F(D+1)

One of the most stunning simple results in all of mathematics.

Notice you get back the reduced binary quadratic form! So it is iterative.

And in fact, shows you how in general to connect a binary quadratic form to an

*infinity*of other binary quadratic forms.

So there is in general this infinite web of binary quadratic Diophantine equations--linked together. Its impact reaches across vast stretches of number theory. I ponder that result and ponder.

That is found from my method for reducing binary quadratic Diophantine equations.

That beautiful simple relation is used by me quite a bit and you can see it pop up in a lot of my more popular posts!

Its power and reach are vast but it can be demonstrated simply with well-known equations.

Plugging in values, let F = 1, u=x, v=y, and D = -2.

Start: x

^{2}- 2y

^{2}= 1

Iterations:

1. (x+2y)

^{2}- 2(x+y)

^{2}= -1

2. (3x+4y)

^{2}- 2(2x + 3y)

^{2}= 1

3. (7x + 10y)

^{2}- 2(5x + 7y)

^{2}= -1

4. (17x + 24y)

^{2}- 2(12x + 17y)

^{2}= 1

and you can keep going out to infinity, but I'll stop with 4 iterations.

Notice you get solutions with x=1 and y = 0:

Start: 1

^{2}= 1

1. (1)

^{2}- 2(1)

^{2}= -1

2. (3)

^{2}- 2(2)

^{2}= 1

3. (7)

^{2}- 2(5)

^{2}= -1

4. (17)

^{2}- 2(12)

^{2}= 1

The solidity of exactness is so comforting. Perfection.

You can actually verify the result by just multiplying out and simplifying:

(u-Dv)

^{2}+ D(u+v)

^{2}= F(D+1)

Which is kind of fun, and if you do it, you'll find it just reduces back to the original:

u

^{2}+ Dv

^{2}= F

So if you run into someone trying to convince you negative things about me you can challenge them to explain the above, and you can wonder why it's not in mainstream literature.

I know I do.

Criticism is easy--people can

*say*anything. But the math does not lie.

Want to see it used with something more complicated? Then check out the link below:

Connecting Diophantine hyperbola to ellipse

James Harris