Simply more integer-like numbers

Came across a rather basic proof using simple algebra, relying on quadratics, which shows there are some new numbers which are really cool. Have a post with a proof which will use as a reference in a more general explanation here.

First comfort is will be in complex plane and considering:

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.

That is a generalized quadratic factorization, where IS important functions are normalized. Which just means equals 0 at x = 0. And is deliberately asymmetrical, where I need to remove that to find solutions, but will do something clever, and introduce k, along with new functions.

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

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

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)

For here relaxed a requirement that is in the proof, as imagine k = 1 or -1. Then is really just a symbol, not doing anything, but also notice NOW we have symmetry. And I can solve for the f's as roots of the same quadratic, which I can make sure has integer coefficients.

So I can force the f's to be algebraic integers. THAT is way important. With one more function have a handle on things, and made it H(x) for that reason. That is for handle.

And introduce H(x), where I like using the capital letter here for visual reasons, but it is not to signify H(x) must be a polynomial, where:

f₁(x) + f₂(x) = H(x)

Where yeah, next you just substitute for one of the f's and simplify sensibly. Is VERY easy algebra folks. Hard to imagine how could be much more simple. And get:

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

So yeah algebra is SO EASY which is good, to remove reasonable doubt! As believe me we are shaking the foundations of number theory. Can you feel it? I can.

For one of my favorite examples have H(x) = 5(7x-1), where have a post where give here. See that yeah, if you have x be integers, then the f's are going to be roots of the same quadratic with integer coefficients which is VERY important.

With k = 1 or -1, we now have a way to find algebraic integer solutions for the original factorization! Way cool. And in fact, with that handle H(x) we can get every possible one, which is where things would end, if we didn't have that clever k to let us do something different.

If we consider k, NOT a unit, so is like k = 2 or even better k = 3, or k = 4, then remember we have:

 g₁(x) = f₁(x)/k

Which is just algebra.

P(x) = (g₁(x) + 1)(g₂(x) + 2)

And we now know that one of the g's cannot be an algebraic integer when k is not a unit, for non-rational solutions for the f's where show that in the proof that they must be there. We can force that factor. The algebra doesn't care. So humans thought something wrong for over a hundred years? The algebra is the reality.

Human beings? Can believe anything.

You may wonder, what if we FORCE the g's to be algebraic integers, like take one of our solutions with k = 1 or -1 when we can get all such possible, and THEN do the symmetry thing as well?

The algebra doesn't care, but will NOT let the f's be roots of the same quadratic with integer coefficients then with k not 1 or -1. They will disjoint from each other. What is key is the process where they are roots of the same quadratic with integer coefficients. As far as the math is concerned, you simply have one of the f's multiplied by k, so its removal to get one of the g's is as easy here as anywhere else.

For human beings though may be harder to process what is harder to see direct. The logic is straightforward, but human emotion can get in the way. It can feel off. Which is something physics students face with quantum mechanics worth noting. Your feelings? Don't matter. Logical rules. Do.

Which goes to show you how mathematics can be EASY and very logical, without question where every possibility has an answer. The problem here then with acceptance of the result is entirely social, as it does not get much easier than a quadratic.

These numbers are also integer-like, which means they are like 1+5i which is a gaussian integer, in behaving in ways very much like integers, while being non-rational.

A way more wordy post goes into lots more talk around it all and can be found here. But for example with this approach can prove that 3+sqrt(-26) actually has 7 as a factor for one of its solutions.

Which one? We can't tell. Like 1+sqrt(4) = 3 or -1. Convention can say take the positive, but the math does not care what you, just some human, thinks is ok. The math knows that sqrt(4) is 2 or -2.

So we cannot see direct when can't resolve the square root, and the math doesn't care about that either.

So yeah, these numbers have weird properties too. Which makes sense. Helped them stay hidden until a smart argument looking for them was found. And should be exciting, but number theory operated for over a century with folks not realizing they were there, and social consequences in our time are HUGE.

But the math does not care.

And yeah, did NOT just figure this approach out at random. Was looking for it, thanks to a published proof.

Have questions? Ask away in comments please.


James Harris