Some abstract reductionism demonstrated

With Diophantine equations in two variables you can see a LOT of apparent complexity with:

c₁x² + c₂xy + c₃y² = c₄ + c₅x + c₆y

Here x and y are the unknowns to be figured out, of course. But it turns out you can reduce which was known, but my abstract reduction methods make that easier in general to get to the basic form:

u² + Dv² = C

And with more generality than with other methods derived by Gauss.

But on that form those same methods--iterate:

u² + Dv² = C

then it must also be true that

(u-Dv)² + D(u+v)² = C(D+1)

___________________________

And I decided to call that a binary quadratic Diophantine iterator, or BQD Iterator for short, and I find it fascinating. It stands alone in that you CAN verify it simply by multiplying out. And conceivably could have been discovered just by someone playing around but apparently was not.

Where yeah there are implications then of course for integer factorization too, which I usually don't like to mention, as EVERY integer factorization can be connected to an infinity of others, trivially.

So now we know that the very complex looking expression, with integers, connects to very simple, and more formalized abstract reductionism makes that so much easier--was able to improve upon Gauss.

And how many people can say THAT?

I've played a lot with it, and you can see plenty of posts by clicking on the label below.

That is just my favorite demonstration example of SOME abstract reductionism. Have pondered it for years. Why can find something one way which had NEVER been found by another? Maybe does help a lot when is EASIER from a formalized process. 

The point important to make then, being abstracted by a process, which I have used in more than one area. Just really easy to see quickly with certain things.



James Harris