The Collatz Conjecture is fun for those of us who have never taken mathematics in university in any meaningful way. It's also probably fun for all of you who have gone to university.
So I want to ask about some basic relationships about the Collatz Conjecture.
For some reason $3n+1$ and $n+1$ work on the odd edge and no others. It's easy to figure out why these do not ever run into infinity unless you start from infinity.
Generalised, given n(k+1) → if odd: $a × n(k) + b$; if even: $n/c$,
It's also easy to figure out that in $mod(ac)$ that each modular instance excluded from $mod(a)$ and $mod(c)$ should point directly towards a power of 2. In $n+1$, 1 → 2. In $3n+1$, 1 → 4 and 5 → 16. Any subsequent odd will be found as members of these excluded modular instances.
Now the funny thing about these two is the odd edges. If you take out the even edge, the instance $n+1$ will have a converging ratio of $\phi$ (Golden Ratio) between any $k$ and $k+1$. The instance $3n+1$ will have a converging ratio of $\phi(3)$ (Bronze Ratio).
Now I know that Bronze Ratio is related to the Heptagon (the longest diagonal divided by the shortest edge will equal $\phi(3)$. And there is a whole bunch of exciting stuff in this article that is a bit above my head. https://en.wikipedia.org/wiki/Heptagonal_triangle
The satisfaction of the cubic mentioned in the article looks interesting (although it might be uninteresting, I have no idea.)
Now, my math is not at all advanced enough to cement the connection here. As I said, I didn't take math beyond high school (Financial math is not enough. sigh.) But my spider senses are tingling here. Is anyone else's spider senses tingling?
Edit: I have edited this question to be as clear as I can muster. I had an error in which identity I was drawing attention to but have corrected that (I hope.) What I am looking for is whether $\phi$ and $\phi(3)$ are of especial interest in the Collatz conjecture.
Well, I gave it a whirl. The result is surprising.
The harmonic mean of $n$ and$\phi_3$ seems to be the experimentally obtained number of "somewhere around $\frac{3}{4}$." Or that was how it was described on Wikipedia.
That is:
$$\frac{Decay}{Step}=\frac{\phi_3}{1+\phi_3}\approx0.767...$$
I predict in a non-canonic fashion that for any a and b, we can obtain a Pisot number, $\phi_{a,b}$ to derive a decay rate per step.
$$\kappa_{a,b}=\frac{\phi_{a,b}}{1+\phi_{a,b}}$$
Where:
$$\phi_{a,b}=\frac{a\pm\sqrt{a^2+4b}}{2}$$
However, this does not translate to heptagonal triangles under any elementary euclidean measure. I can't say if anything is revealed in hyperbolic geometry because I have no way of working with that stuff yet.