I have a few interesting hypotheses (NOT trying to prove the conjecture!!), if anyone is willing to provide some advice or assistance?
I am admittedly not knowledgeable of most higher order mathematics, nor most Collatz literature. However, I do play around with Collatz, NOT to try to prove it, but to see what sort of interesting things pop up to me. I ask for help here because although I have run some of these ideas by AI systems, we know how those can be. And I figure actual, real mathematicians and people who are deeper into the Collatz world would be more beneficial.
I have a couple of hypotheses I've been working through (they're mostly empirical/heuristic/something weird I noticed that you all probably already know), and if there's anything novel here, I'd appreciate some guidance.
Ok, here we go:
My first one is what I call the "Dependency Loop". The best way I can describe it is this way: The difficulty in a proof is that all known attack vectors return to depending upon trajectory analysis.
I also have one I refer to as the "Null Tautology" which is very similar to the dependency loop, but I describe it like this: candidate sufficient conditions for the conjecture often seem to require proof effort comparable to the conjecture itself.
Which leads me to what I call the "Principle of Conservation of Difficulty". Which basically says that reformulating Collatz preserves its difficulty and complexity.
Which leads me to another hypothesis I had the other day. Examination of "prime chains" in the conjecture itself.
Like the first few numbers look like this:
3: 5
7: 11, 17, 13, 5
9: 7, 11, 17, 13, 5
11: 17, 13, 5
13: 5
With the last prime in the sequence being the "terminal prime". My hypothesis here is this: Hitting the first prime in the sequence "chains" you to other primes that are "chained" to a "terminal prime". Thus hitting the FIRST prime "chains" you to a terminus. The issue there, it would seem, would be showing that all trajectories have a prime number (which we don't know, and seems hard to prove).
And from that, I started breaking down trajectories along the odd map into what I call "micro-trajectories", which are shorter trajectories from one prime to another. Think about it like the Odd-Odd map, but with primes, and the "in between" calculations are the "micro-trajectories".
Here's an example using 9's full trajectory:
28, 14, 7 (first micro-trajectory ends here)
22, 11 (second micro-trajectory ends here)
34, 17 (third micro-trajectory ends here)
52, 26, 13 (fourth micro-trajectory ends here)
40, 20, 10, 5 (fifth micro-trajectory ends here; 5 is the terminal prime in this trajectory).
Just some thoughts I had. Can anyone shed any light on these? Are any of them interesting or worth pursuing at all, from a research standpoint? Again, NOT trying to prove the conjecture, but it would be interesting to add something to the literature that might be useful to someone else.
Go easy on me. ;-)