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Thursday, January 25, 2018

Probing number properties with quadratic factorization

Feel extremely lucky that a simple quadratic factorization was available for checking a number of things, and was around 2010 started studying. Will talk how that basic factorization allows probing deeply into numbers.

In the complex plane:

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

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

And is VERY important that the g's are normalized. While studying with algebraic integer solutions may have been done, came up with a generalized way to do that--and beyond.

Introduce k, where k is a nonzero, and new functions f1(x), and f2(x), where:

g1(x) = f1(x)/k and g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

And introduce H(x), where: f1(x) + f2(x) = H(x)

Then just solve for one of the f's, and substitute so I can find:

f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0

Which gives me total power to force a monic quadratic with integer coefficients by how I choose k and H(x). And then you can solve for f1(x) using the quadratic formula:

f1(x) = (H(x) +/- sqrt[(H(x) + 2k)2 - 4k*P(x)])/2

And H(x) is a handle for every possible factorization with the g's, with k = 1 or -1, you just get algebraic integers if you have H(x) be a polynomial with integer coefficients.

However, you now have that if k is some other nonzero integers besides 1 or -1, then one of the g's can be blocked from being an algebraic integer by: g1(x) = f1(x)/k

That blocking of course occurs when non-rational.

But the product then is still allowed to be an algebraic integer: f1(x) = kg1(x) 

The entanglement with non-rationals is FASCINATING. And yet, k is proven to be a factor.

Here should note is where you can introduce contradiction! If instead of declaring in the complex plane, could get contradiction or not if declared in ring of algebraic integers just by choice of k.

The main point there is simple: one of the g's provably cannot be an algebraic integer if nonzero integer k is not 1 or -1, and the g's are not rational, when H(x) is a polynomial with integer coefficients. While one of the g's IS being forced to be an algebraic integer.

Where simplicity in the proof comes from the f's being roots of the same quadratic. H(x) is key here. Depending on how is chosen you are then pushing back to pick the g's.

Which is how the analyst is choosing from infinite possibility.

Should admit was EXTREMELY surprised that the algebra DOES recognize as distinct roots of monic polynomials with integer coefficients i.e. algebraic integers.

And did not fully appreciate against my biases until this post November 2015, when finally considered k = 1 or -1. Where had been considering algebraic integers as just kind of a mistake since my own research from 2003, so had around 12 years with that idea in mind. But the math actually has them as distinct and is curious in what ways versus other members of my ring of objects.

Where at least know that there are other members NOT algebraic integers where multiplying times some non-unit, nonzero integer gives a product that is.

Oh so yeah, bringing up my ring of objects which includes all integer-like numbers as well as integers themselves. And my ring of objects excludes any numbers not integer or integer-like, which was the approach I used to include them all. Switching to abstraction of exclusion to include versus attempting to get all with rules of inclusion.

So much from a basic factorization! And also feel lucky and amazed with the power of abstraction shown with this approach! Finding is one thing, but is great for that mathematical logic to be there.

These are mathematical pieces giving a handle on some aspect of infinity.

That is wild. To me H(x) is fascinating with that ability to handle. It is meant to cover an infinite number of possible expressions where as a human researcher am focused very narrowly. Getting a handle on infinity in this way intrigues me as a powerful technique.

Having a quadratic factorization available then leads to an easy way to thoroughly check on number theoretic properties.

Reference post: Simple Generalized Quadratic Factorization


James Harris

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