Tuesday, January 20, 2015

Unfortunate number theory coincidence?

Number theory deals with integers, like 1, 2 and 3, or -1, -2, and -3, and of course 0. But for over a century it has also been believed to also have some relationship to complex radicals, like sqrt(-D),  like when considering:

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.

James Harris
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