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Friday, November 13, 2015

Such a simple thing to try

One of my favorite simple things to suggest to those mathematically inclined involves a basic algebraic factorization:

Consider:

k*P(x) = k*(g1(x) + 1)(g2(x) + 2)

where P(x) is a quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x. And k is any nonzero non-unit integer.

There is a trained habit to toss away extraneous factors like k which may be why something really actually kind of strange was missed! Great for me! Such habits can open the door to a person who simply breaks them, as you can pull that k in easily enough with some manipulations and new functions and have:

k*P(x) =  (f1(x) + k)(f2(x) + k)

One way of looking at it is that the k is now entangled, which is why it isn't to be a unit. And the reason I say it's entangled is that NOW it can be much more complicated to try and remove it, depending on the functions.

Because now the right is symmetrical, allowing me to bring in algebraic integer functions, but the left is still asymmetrical, and this forced symmetry breaking lets you do some weird things.

Here's an example with actual numbers:

7*162 = (5(3+sqrt(-26)) + 7)(5(3-sqrt(-26)) + 7)

Forced symmetry breaking is a concept I think that should be taught. Just DO it.

Why? Because with that symmetry the f's can be algebraic integers, but one is forced by this construction to have k as a factor, which is a neat trick! It leads to a discovery of numbers where ANY k with the rules will make one of them an algebraic integer, when it's not if not multiplied by k.

That is, consider such a number I call an object o, which is NOT an algebraic integer. It can be proven, rather easily actually, that for any k, an integer not zero or a unit, that k*o is an algebraic integer. That's weird.

If you wonder why, consider some number x which is NOT an algebraic integer where multiplying times 3 makes it one. An example is x = 1/3. But notice that multiplying by 5 does NOT make it an algebraic integer as then 5x = 5/3. But these odd numbers don't care. If you multiply by 3? Yup, will be an algebraic integer, but ALSO if you multiply same number by 5, will be too! Or any of an infinity of other integers.

Proof is trivial. Requires nothing but elementary methods as in basic algebra.

But to get the result you have to DO something very much against the trained instincts of most in mathematics I'm sure, as one of the earliest things taught is to throw away extraneous factors NOT to add them!

Here's a reference to full post:

somemath.blogspot.com/2015/04/generalizing-non-polynomial.html

Breaking symmetry in mathematics seems to be something I like to do! And to me it's such a simple thing to try. Maybe it's the rebel in me.

This result endlessly fascinates me and have another post summarizing conclusions:

somemath.blogspot.com/2015/04/summarizing-non-polynomial.html

These special numbers I found are so AWESOME. So I kind of feel funny talking about them as weird and reality is that for years now have felt a special connection to them. They have always been there of course. But now out in the open in our reality they could explain so much.

You see, if they are relevant to our world they always have been, possibly controlling our reality or helping its rules in ways we simply could not imagine, because we just didn't know they existed.


James Harris
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