So I must take my time and accept great difficulty in establishing unimpeachable certainty. Which is fun! It pushes me to understand my own research and has even aided further discovery. But also there is so much demonstrative now, done.

One of those ideas that demonstrates has to do with Diophantine equations like:

**c**

_{1}x^{2}+ c_{2}xy + c_{3}y^{2}= c_{4}+ c_{5}x + c_{6}yWhere is binary because you have unknowns x and y, and is also called two-variable. Diophantine just means looking for integer solution. I figured out a best way to reduce to something simple.

For example:

**x**

^{2}+ 2xy + 3y^{2}= 4 + 5x + 6yWhere I just tried something simple and was lucky there ARE integer solutions. My method for reducing gives:

(-4(x+y) + 10)

^{2}+ 2s

^{2}= 166

And turns out you can easily solve from there and find integer solutions:

**x = 4**,

**y = -2**, or

**x = 5**,

**y = -2**

More recently played around and had to work harder for this example, as wanting something simpler in ways:

**x**

^{2}+ y^{2}= xy + x+y + 102Which my method reduces to give:

(-3(x+y) + 6)

^{2}+ 3s

^{2}= 3708

And figured out solutions:

**x = -10**, so

**y = -8**, or

**x = 3**,

**y = -8**

Copying from prior posts.

Reference posts: Reducing binary quadratic Diophantines, Reducing a quadratic Diophantine to find solutions

My method for reducing is I think the best in the world, and supersedes methods that trace back to Carl Gauss who is a HUGE hero of mine. And his methods now include wasted effort finding something called a discriminant, which is not needed for reducing these equations.

People wasting their mental energy though is not surprising to me, or a great concern. Others might value their efforts more highly than I do, as is a telling failure, in my opinion.

From my perspective, is more telling when people work harder than necessary, when they could do better than those who find better.

Those who seek truth best, interest me more than those who do not.

May seem cold, but many may wish to be something they are not. To me is important how to figure out those with potential. Finding the truth when is so easily available? When easy to find on the web?

If you can't do that easy then how can you find the things waiting to be discovered, when that can be so hard?

Challenges are met by people with ability. Why I deeply appreciate the challenges in front of me, as I learn so much more about what I can do.

Everything testable and each statement of mine checkable though I wonder if am reaching in claiming is the best possible. But I think not. Like consider, how I got something important.

Use my method to reduce on: x

^{2}- Dy

^{2}= F

Gives the BQD Iterator.

James Harris

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