BQD Iterator is a very surprising tool

Realized that one of the more profound results for me both practically and philosophically is a mathematical tool I call the Binary Quadratic Diophantine iterator, or BQD Iterator for short.

Where have this post as my established reference, posted November 28, 2014, which talks in-depth. For here will just give my favorite form again.

Given: u² + Dv² = F

then it must also be true that

(u-Dv)² + D(u+v)² = F(D+1)

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Where do minor variations on that theme here and there as of course can shift with symbols used and signs, and what is weird about that thing is, seems to connect to just about every property of integers.

Wouldn't that be one of the weirdest things ever? Like, ONE thing being key to so much where for thousands of years so many searched and found pieces of the puzzle, when the biggest thing was SO simple.

And yeah I know, may not look like much. Have stared at it often for years now, pondering.

The BQD Iterator quite simply may control, or in so way be related to the controls for all integers. It is the one tool that the math may use, just about everywhere with integers. I find that remarkable, and wonder if is true.

Like MAY lead to THE template for ALL integer factorizations: But does it factor?

Where is one of those areas where I do NOT check thoroughly, as delves into too much scary. (And will collect here another post where talked scary areas with this post with regard to implications from other things.)

The AMAZING thing about the BQD Iterator is that no human found it directly, which includes me, as I used tautological spaces. What is it about it that not even just playing around no human just stumbled across it? Or maybe someone did, but is just not big in the historical record? Have wondered.

And I LOVE to play with it now. Though took a few years, where helped out much when named it for some reason. Like sqrt(3) approximately equals x_n+1/y_n+1, where:

x_n+1 = 1351x_n + 2340y_n

y_n+1 = 780x_n + 1351y_n

and x₀ = 1, and y₀ = 0;

So get x₁ = 1351, and y₁ = 780. Which is a decent approximation, and next is:

x₂ = 3650401 and y₂ = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Have used to explain result where Euler and Ramanujan played around, along with lots of other things. And emphasized for me just how much simple we may never know, as how do we know what we do not know?

So much of being human is about aspects of our design with which we are simply born, which must impact much what we can discover. Which is kind of depressing actually.

And yes, talked in this post where mused about my recent find of a third primary way to calculate the modular inverse. Where have been trumpeting that one for a bit now.

For me it is interesting then to have a result where yeah, was me there. And the math didn't hand me THAT result, but then again took me years to notice something that followed from something clever. Where kind of wonder, why did I think of it?

But then again, have been as I say, talking to the math now for quite some time. And has dawned on me that I DO think differently now. Maybe there does happen that better ability to look outside the box as the saying goes, when have watched a process give answers WAY outside of what humanity found on its own.

Yeah that does make sense. Watching the math DO math, could conceivably have shifted how I look at mathematical problems. Like if had watched some human teacher working on math, but am watching an infinite intelligence instead. Wow. These are the kind of posts where am glad just start typing and find out where will go.

Like, what greater teacher of mathematics could there be to watch? Than the math? As can watch the math DOING mathematics, and just be like a student, learning?

And later, me, the student could figure out something cool entirely on my own. Yeah. Maybe.

So yeah, watching the math, do math, can help teach a human, me, how to do math better? Wow. What a concept. Can explain much if true. And I do so like to hedge, do notice. If true.

Yeah but did have the packing of spheres thing, before. Though still bothers me for some reason, how simple that modular approach is. And of course, did figure out myself how to use tautological spaces. Still guess do love the idea of the math itself as one of my greatest teachers. And why wouldn't I?

Makes sense to me.


James Harris