For example, if you dare, in the complex plane 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.
Yes, I know the k can be removed, but now see what I do as instead I use it to allow me to force symmetry on the right.
k*P(x) = (f1(x) + k)(f2(x) + k)
where: f1(0) = 0, f2(0) = -k+2
And now the k is wrapped up! How do you remove now?
There ARE simple solutions, for example, try: g1(x) = x and g2(x) = x
That forces P(x) = x2 + 3x + 2
Which means, switchable by indices of course:
f1(x) = kx and f2(x) = x - (k-2)
Which is the simplest case of it. Polynomial cases are easy. But can you go beyond?
If you DO try and get stuck I have answers. And guess what? The absolute truth requires only algebra. And understanding that truth requires that you can step through a mathematical argument and care fiercely about wanting the truth. Because the typical human mind may balk at trying to go beyond polynomials.
I figured out a way to get a handle on the equations with a function H(x), where yeah the H is about "handle", and I capitalize because I like the look, though actually it can be an infinity of functions in the complex plane:
f1(x) + f2(x) = H(x)
So now I can substitute out for one of the f's, and choose easily:
f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0
Which covers an infinite class of equations now. Easy.
Now we can see the obvious, but also see how you step off into beyond. And notice that all I did was do something counter to habit. That is easier said than done.
So many talk about stepping outside the box, but the point is that thinking a certain way is easier than not.
That I have a quadratic isn't a surprise of course, but now can use the quadratic formula which has a square root! And that has HUGE consequences.
And you can solve for f1(x) using the quadratic formula:
f1(x) = (H(x) +/- sqrt[(H(x) + 2k)2 - 4k*P(x)])/2
Because H(x) is actually a function with x as a variable, turns out that to get that square root to go away, you need a quadratic. And that means that then the f's are just simple polynomials themselves, linear.
There is no way around it.
Also if H(x) is itself an algebraic integer function, that is, give algebraic integers for algebraic integer x, then know that the f's are as well.
Weirdly enough, H(x) can cover ALL possible solutions for the g's and f's, but is limited with my proof by: H(0) = -k + 2
That just amazes me. It in essence tells the math all it needs to know about what I want, as I picked the initial conditions. And wanted 1 and 2 in there for simplicity.
Here's something with actual numbers:
7*162 = (5(3+sqrt(-26)) + 7)(5(3-sqrt(-26)) + 7)
Ask yourself, what are the values for the g's here?
And I've talked about this subject quite a bit including a recent mathematical proof. While I've also noted that the first post on this blog back in 2005 relates to this subject as I've been considering mathematics in this area now for over a decade.
LOTS of controversy in this subject area as well, including a dead mathematical journal no less.
These kinds of things are fun, weird, and fascinating as in human history they don't happen that often.
People living through such things? Rarely have a clue how huge it all is. Which probably has its benefits I guess. Maybe I shouldn't tell you then, eh? But I'm not worried, only a special group of you will believe me early anyway, I'm sure.
At least at first, while eventually of course will just be settled mathematical history.
That weird feeling though that some may have with the result and its consequences is all about limitations of the normal. We think a certain way! It can be SO HARD to shift even with absolute proof.
Am guessing the human brain wasn't built for such mathematics as I've discovered, as even for me has taken some time to get comfortable with it. But the great thing is with effort our minds can comprehend the math anyway.
Makes me question how any people could think that human beings create mathematics rather than discover it, which is why an ego check like this result may have such a hard way to go! After all, for some self-important person that may be a cherished belief.
So maybe for some people it's a welcome to humility if they accept it.
Welcome to mathematical reality where you're small. The Math is infinite.
And guess what? The Math does not need you. It does not need you to understand.
The Math does not care.