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Wednesday, November 08, 2017

Easily explaining a historical miss

They were trying to solidify the foundations of mathematics as a discipline, in the late 1800's. Gauss was already gone. One of the most influential mathematicians of all-time, he had died in 1855. Yet one of the areas they felt had to be addressed as they tried to handle all numbers were what we call gaussian integers in his honor. Numbers like 1-3i where note (1-3i)(1+3i) = 10, behave a LOT like integers, so were considered integer-like while not being rational.

Those mathematicians working to turn mathematics into a formal discipline seized upon roots of polynomials. Considering roots of monic polynomials, meaning lead coefficient was 1 or -1, with integer coefficients seemed to work! These roots encompassed gaussian integers and more, and they tried various tests where things seemed ok. They decided had the ring on which number theory could rest, which they called ring of algebraic integers. Found a link to Wikipedia on algebraic integer where oddly enough they don't talk the history. Algebraic numbers are a ratio of algebraic integers.

What we know now that those mathematicians apparently did not, is that a problem can be shown with a generalized quadratic factorization.

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.

But to show the problem you have to force symmetry, but also do something unintuitive and ADD an extra variable. That is actually counter to how math students are taught to go, as training is to remove extraneous variables, not introduce them!

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)

The purpose of forcing symmetry is to allow for solutions where also need one more function, then can solve for the f's using the quadratic formula.

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

Where easy algebra allows for me to solve for the f's using 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, and has a key constraint: H(0) = -k + 2.

Copying from post: Simple Generalized Quadratic Factorization

If we let our extraneous factor k equal 1 or -1, no problems, and you get solutions that are roots of monic polynomials with integer coefficients when use integers for x and have P(x) and H(x) with integer coefficients. But let k equal some other integer, and one of the g's must be out of the ring as:

g1(x) = f1(x)/k

Which can show with an example. 

Let P(x) = 175x2 - 15x + 2, H(x) = 5(7x - 1), and k = 7.

Plug everything in, and you can solve for the f's, and with x=1, get an interesting conclusion, as then you find that one of the solutions for 3-sqrt(-26) has 7 as a factor.

Have ran into problems where people learning a convention with square roots can see that as a single number. But note that 1+sqrt(4) = 3 or -1. So is improper to say it has 3 as a factor, while one of its solutions does.

That we cannot see the factor of 7, with 3-sqrt(-26), does not change what is mathematically proven with our generalized quadratic factorization. However, it is a standard result in number theory that cannot have a non-unit integer as factor in the ring of algebraic integers.

It is hard to imagine those mathematicians over a hundred years ago coming across such an argument to find there was a problem with their attempts at formalizing sturdy foundations for mathematics as a discipline.

Today we know far more, especially about impact of over a hundred years with this result, as number theory became the biggest draw for people entering the field, where I think is because was easier. Is a quirk of mathematical fallacy. Results in number theory went further and further away from integers, where is now dominated by complex analysis, ever more abstruse, and much of it? May simply be wrong, but how would you know?

As a discoverer came across this problem with another argument where got a paper published in a dramatic story, as I used the problem to show one result by the rules, which could also be shown wrong by those rules. That is, I proved a result with the ring of algebraic integers declared, used valid expressions and operations in that ring, yet got a conclusion not true in that ring.

What happened then was telling. 

Have talked it in detail in this post: Publishing a contradiction

A math journal possibly being brave published my paper, then a chief editor tried to run from it, and later the entire journal shutdown.

Given that was over 13 years ago, best guess as to what was the purpose? Is what was obtained: millions of dollars in continuing funding primarily from governments for number theory research which may simply be bogus.

Such a simple thing in trying to establish solidity to mathematics in simplicity of purpose. Such a difficult thing to pull off in actuality. Cannot fault their motivations. Their lack of ability was unfortunate. One wonders if Gauss had still been alive would he have quietly figured out there was a problem? Don't want to take hero-worship too far, but his brilliance was demonstrated and his gaussian integers do not have a problem.

Who knows.

What we do know is that in running from the error in our times, which thankfully have learned to demonstrate more easily now with my generalized quadratic factorization, is that fraud took over number theory research. Motive? Money, status and false recognition.

There is no other conclusion available. The math is simple enough.

Scientific progress and technology were protected by relying on applied mathematics. Which is great!

But for the error to be fixed requires acceptance of pure math research.

And the bravery to accept mathematical truth, in order to once again have number theory focused on the pure pursuit of knowledge.


James Harris

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