Woah

Okay, so, recently I got a little interested in the Collatz Conjecture.
I wanted to try and approach this problem from a different perspective than a lot of the literature seems to be on. What I really wanted to do was approach this problem as a chaos theory problem, since that branch of mathematics is concerned with maps on itself. But I realized that I just needed a function that I could map to itself. I decided to define a function C(n) such that

where
g(n) and h(n) are rather unremarkable terms here. They are terms that make C(n) satisfy the requirements

and

While g(n) and h(n) may be unremarkable, E(n) is very remarkable indeed. E(n) is a function that I dubbed the even function. It returns +1 if a function is even and -1 if a function is odd. This leads to some very interesting cases.
First of, I realized that there are three distinct classes of numbers:
Numbers that are powers of two, and when divided (n - 1) times by two will remain even (but when divided n times become odd):

Numbers that only have two as a single factor:

And finally, odd numbers:

The behavior of all of these sets of numbers is well defined in the even function. For instance, the first two classes will always return +1. The third class will always return -1, by definition. I wanted to take it a single step further, however. And so, I asked myself: what about non integer numbers? What about things like 1/2? Well, I thought of a clever way to derive that.
Because this is the even function, for two numbers ,
which you can easily verify. With that in mind, I looked toward the half integers. The half integers can be defined by

Now, taking our identity from a second ago, I know that

which means that

which finally means that

This can be expanded further for all 2^n type fractional integers, and all it does is keep adding more roots like so

It took me an inordinate amount of time to reach this point, and I was blown away when I reached that conclusion. I'm not a professional mathematician, and so I've kind of just been trying stuff in my free time. But today, I finally came up with this method for finding the "evenness" of half integers. I don't think this method is really meaningful for other primes, as all other primes are odd, and naturally return -1. As for what this has to do with the Collatz Conjecture, I think that defining the function for all numbers (real and imaginary) will make it easier to solve, because with the definition for the function I gave, it's simply a problem of proving an attractor at near 1.
So, what do you think. Kinda cool, eh?

Comments

[Lazauya]

Hey! Note to anymore who's reading this: you might need to reload a few times to see all the math, because they are rendered on another (very slow) server in real time. The images come, you just need to wait a sec! Thanks.