x2 - Dy2 = 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: x2 - Dy2 = 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) = x2 + Dy2 - 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 x2 + Dy2 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.