Like in accepted mathematical training students know that you can have a unit paired with a non-unit with integers, for example:
x2 + 4x + 3 = (x+3)(x+1)
However, it is also taught that goes away if NOT rational, and math people teach that with non-rationals you can only have units that remain integer-like, with something like:
x2 + 4x + 1
Where does not factor easily like before. Which is just given as a property of numbers not integers, different from numbers that are integers, when I've proven is a property of the ring of algebraic integers.
Like if I say evens, and then claim that 2 and 6 do not have factors, in evens because 3 is odd. Where evens are not a ring, but is a ready example to highlight how you can easily see a case, where you get something weird. Explanation? Supposedly non-rationals are just different from integers, even when integer-like, as algebraic integers are.
(Oh yeah, so that's NOT a real rule for numbers. With a proper ring, yes, a unit in that ring can always be paired with a non-unit and both be roots of a monic polynomial with integer coefficients. So NO distinction between integers and other integer-like numbers there.)
The proof that there are actually other numbers along with algebraic integers though, just like there are odds along with evens is so easy, I prove it with elementary algebra and also had a prior result related published in a formally peer reviewed mathematical journal, where there is a wild story with the chief editor trying to delete out of the electronic journal after publication and later the journal died.
Also, have piled up mathematical results, including my most recent which is a new way to calculate the modular inverse, which seems built for the computer age as is iterative in a way that works well with computers, so seems perfectly timed for our now. That actually puzzles me a bit. Seems so perfectly timed. Well is here now. And that was months ago.
Takes a LOT to try to ignore something on that level. Humanity only gets so many basic discoveries of that type in ITS entire existence, am sure. There are only so many fundamental results.
Why does the proof of more integer-like numbers impact number theorists in an important way?
Because number theorists are the ones who may operate with ring of algebraic integers, while others, like scientists as also consider beyond mathematicians, tend to use mathematics in fields.
There is no problem in fields. Like notice, if you have 6/2 or 2/6, in a field you can divide through with no problem. As is only with an arbitrary rule where factors matter that you have a problem, like saying only evens.
With ring of algebraic integers, you are saying, only numbers that are roots of some monic polynomial with integer coefficients.
That restriction is arbitrary like saying evens is arbitrary, and the algebra no more recognizes it, than not letting you divide 6 by 2 because you say evens.
So the mathematics is EASY. It is easily proven. And the problem is easily explained, yet over a decade has gone by, and we're past the point of benefit of the doubt for number theorists. Oh yeah and that weirdness with the math journal. Seems like lots of deliberate action.
The best explanation for trying to ignore such a result?
Seems that the simplest explanation is that enough number theorists have research, possibly on which they have based their careers, where it turns out the truth would invalidate that research.
(Do you have a better one? If so, please comment. I'd LOVE to see it.)
Here's the weird thing: if that's true then number theory WOULD crowd. That is, the path into mathematics through number theory would be wider because of the error. And EASE of success would be greater as well. Of course problem is, if try to show that success with actual integers! Unless you tended to avoid them. And greatest success in number theory?
Would probably be for people who MOST used the error. And scariest people?
Applied mathematicians. They would remain ERROR FREE and would need to be sidelined in case they noticed the problem. Or began checking number theorists more closely.
Oh, but of course number theorists DO have one applied area. Math involved in much of computer security globally. Like lots with public key encryption. I hesitate in discussing that area. But it definitely gives pause. Especially if you pay attention to news and routine breaches, which somehow always get explained. But for these people? That may just be what they now accept.
There though WAY out of my area. Just seems so impossible. Requires governments all over the world to just not be paying enough attention, if these people have reached that level of faking it.
Hence the social problem. Now how to solve it?
So far they seem in my opinion to be relying on benefit of the doubt. Looking for people to rationalize around the details.
Which must have worked so far as here we are. And I am still explaining something very obvious, where the mathematics is as absolute as ever. And very simple to explain.
James Harris
x2 + 4x + 1
Where does not factor easily like before. Which is just given as a property of numbers not integers, different from numbers that are integers, when I've proven is a property of the ring of algebraic integers.
Like if I say evens, and then claim that 2 and 6 do not have factors, in evens because 3 is odd. Where evens are not a ring, but is a ready example to highlight how you can easily see a case, where you get something weird. Explanation? Supposedly non-rationals are just different from integers, even when integer-like, as algebraic integers are.
(Oh yeah, so that's NOT a real rule for numbers. With a proper ring, yes, a unit in that ring can always be paired with a non-unit and both be roots of a monic polynomial with integer coefficients. So NO distinction between integers and other integer-like numbers there.)
The proof that there are actually other numbers along with algebraic integers though, just like there are odds along with evens is so easy, I prove it with elementary algebra and also had a prior result related published in a formally peer reviewed mathematical journal, where there is a wild story with the chief editor trying to delete out of the electronic journal after publication and later the journal died.
Also, have piled up mathematical results, including my most recent which is a new way to calculate the modular inverse, which seems built for the computer age as is iterative in a way that works well with computers, so seems perfectly timed for our now. That actually puzzles me a bit. Seems so perfectly timed. Well is here now. And that was months ago.
Takes a LOT to try to ignore something on that level. Humanity only gets so many basic discoveries of that type in ITS entire existence, am sure. There are only so many fundamental results.
Why does the proof of more integer-like numbers impact number theorists in an important way?
Because number theorists are the ones who may operate with ring of algebraic integers, while others, like scientists as also consider beyond mathematicians, tend to use mathematics in fields.
There is no problem in fields. Like notice, if you have 6/2 or 2/6, in a field you can divide through with no problem. As is only with an arbitrary rule where factors matter that you have a problem, like saying only evens.
With ring of algebraic integers, you are saying, only numbers that are roots of some monic polynomial with integer coefficients.
That restriction is arbitrary like saying evens is arbitrary, and the algebra no more recognizes it, than not letting you divide 6 by 2 because you say evens.
So the mathematics is EASY. It is easily proven. And the problem is easily explained, yet over a decade has gone by, and we're past the point of benefit of the doubt for number theorists. Oh yeah and that weirdness with the math journal. Seems like lots of deliberate action.
The best explanation for trying to ignore such a result?
Seems that the simplest explanation is that enough number theorists have research, possibly on which they have based their careers, where it turns out the truth would invalidate that research.
(Do you have a better one? If so, please comment. I'd LOVE to see it.)
Here's the weird thing: if that's true then number theory WOULD crowd. That is, the path into mathematics through number theory would be wider because of the error. And EASE of success would be greater as well. Of course problem is, if try to show that success with actual integers! Unless you tended to avoid them. And greatest success in number theory?
Would probably be for people who MOST used the error. And scariest people?
Applied mathematicians. They would remain ERROR FREE and would need to be sidelined in case they noticed the problem. Or began checking number theorists more closely.
Oh, but of course number theorists DO have one applied area. Math involved in much of computer security globally. Like lots with public key encryption. I hesitate in discussing that area. But it definitely gives pause. Especially if you pay attention to news and routine breaches, which somehow always get explained. But for these people? That may just be what they now accept.
There though WAY out of my area. Just seems so impossible. Requires governments all over the world to just not be paying enough attention, if these people have reached that level of faking it.
Hence the social problem. Now how to solve it?
So far they seem in my opinion to be relying on benefit of the doubt. Looking for people to rationalize around the details.
Which must have worked so far as here we are. And I am still explaining something very obvious, where the mathematics is as absolute as ever. And very simple to explain.
James Harris
No comments:
Post a Comment