There are those things which can simply challenge points of view, like one of my favorite discoveries as connects to so much across number theory. Apparently it just connects integers to each other across infinities and with years to ponder, still find myself wondering. Is much about an iteration.
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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My reference source says I found it back in 2008, but just started calling it a Binary Quadratic Diophantine iterator in 2014, where like to call BQD Iterator for short. And it has a unique pedigree, as results from using my method to generally reduce binary quadratic Diophantine equations on:
x2 - Dy2 = F
You switch signs by shifting the sign of D. Where I found out that is the most important expression for integers in number theory, partly because I could generally reduce to it! Where of course can generally reduce:
c1x2 + c2xy + c3y2 = c4 + c5x + c6y
Where am going over old territory quickly as have brought up all before. So yeah, you have this simple relation at the heart of number theory, and can do SO much focused on one expression, like recently found a new primary way to calculate the modular inverse using:
x2 - Dy2 = F mod N
And key here is noticing how discovery propagates. So I finally used tautological spaces against a general binary quadratic Diophantine equation, and found a better method for reducing! Then that method of reducing at some point I used on a fundamental equation and got my BQD Iterator. And then I began to play.
One of my favorite demonstration results like to use lots now to just show:
(462 + 482 + 722)(1722 + 2582 + 4302 + 6022 + 17622) =
6152 + 30752 + 141452 + 159902 + 1884972 = 774*210
Found with it.
Number theory is supposed to have numbers.
The BQD Iterator behaves like the core equation of number theory, clearly commanding much. Still I find I ponder.
Such power in a result draws GLOBAL interest and not surprisingly. And as the years go by will admit have thought more about a responsibility. Should I have it? I say, yes. Discovery doesn't make me but I can feel better that way. The math may not care, but I should. You end up feeling a responsibility for the knowledge which doesn't need me beyond discovering it, but feel like can help others in appreciating properly.
If you do not think a planet of humans notices useful working mathematics then I think that you do not believe in math. Web enables. Knowledge moves very well in our times.
Believing in math is an interesting thing. Either you do, or you do not. Actions tell.
There is a sharp divide there that intrigues me. If you believe it IS the math which is important then focus is there. If you believe actually other things are more important, others can tell.
For me contemplating a global attention reality has been daunting for some time. I wonder how I am to know how, as grew up just outside of Tifton, Georgia, USA. Growing up in a small town rural area of South Georgia has given me much, and I ponder how much works for me now.
It amazes me global attention. Like to talk about the math as like a friend as that is comforting for me, and my imagination makes it feel real. The math is the initial why of the attention and the math stays with me, no matter what.
To my mind when you discover? You present. And let others do THEIR jobs.
Today am a person known around the globe who while struggling with the attention has kept my appreciation for what drives it.
Is interesting that belief in mathematics though, even if does not matter to use. People can rely on math tools without understanding them, or face consequences without believing in them, but some also believe in the beauty of truth. And if you are such a person, is a good thing am sure.
If you believe in math, then you know what discovery can do. Not just for what we know now. But for all our species may ever know.
As someone who is thankful for the knowledge, am working harder to focus on being a responsible bearer of the truth. Besides, it also is SO much fun!
Discovery? Works.
I discovered the BQD Iterator, and am thankful for all it can bring, even though will never know it all. Can just be grateful for more opportunity opened up to our world.
James Harris
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