x

^{2}- Dy^{2}= 1
For years I've pondered something which I think might explain it as a number theory coincidence.

Consider this number theory tool requiring only integers.

Given: x

^{2}- Dy^{2}= F(x+Dy)

^{2}- D(x+y)

^{2}= F(-D+1)

I decided to call it a binary quadratic Diophantine iterator and have an entire post as a reference to it, but you may wonder what does that have to do with sqrt(-D)?

Consider: (x - sqrt(-D)y)(x - sqrt(-D)y) = x

^{2}+ Dy

^{2}- sqrt(-D)(2xy)

It's also iterative, and works a lot like mine if x = y = 1. That square root of a negative number lets you get the positive where x

^{2}+ Dy

^{2}tracks with my x+Dy, and 2xy tracks with my x+y.

With x = y = 1, that's 1+D for both with the former and 2 for both for the latter.

Of course a big difference is that -D+1 that gets stuck on the end with mine.

So far though I've been able to do a lot with no radicals needed. And that's just what I'm figuring out on my own.

I've thought about the similarity noted above for years. Figured I may as well make a post about it, finally.

James Harris

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