[seqfan] Re: This sequence proves the Collatz Conjecture

[Thomas Scheuerle]

Dear Ali, dear all

I think I have understood some small parts of his approach.
Essentially he rediscovered the dynamics of the Collatz function modulo 2^k.
I was forced to look into this topic in some detail for my work in A353063.
A353063 allows to describe the generating function of the sequence 
someone would obtain after k iterations of A014682 applied onto the sequence of natural numbers 1,2,3,... .
In this case the dynamic can be expressed by 2^k states and can be fully recursively calculated from the the result of k-1.
But:
He writes "The second number is d, which is the number of A3 terms that m jumps 
over after one IA operation."
And he writes: "However, in each iteration, d increases compared to its range. For example "
It was hard for me to check all details in his text, but I think I observed something very analogous to his "d" in my work for A353063 too.
The problem is to explain the behavior of this "d" as it points back again to 3x+1 if you try to do so.
It is now difficult to explain what I want to say here, but

@Ali you will start to understand what I mean if you try the following task:
Check what your "d" will do for other functions of the form 3x+m and try to find the explanation why it is special for 3x+1. Or to phrase it differently, why does m jump in 3x+1 always the right amount but not in 3x+m generally ?

I think nothing is wrong to research Collatz deeper as it will enlighten the understanding of complicated dynamical systems. I think this will also make sense without the strict mindset of wanting to proof something.

Personally I have a very heretic opinion, I know that many will disagree with it:
I believe some conjectures are only true by chance or probability.
This means they can never be proven because the law which makes these true contains no absolute statement. There is only some probability of failure with some not absolute convergence (or not even that) to zero. On the other hand I doubt that Collatz is really an example of such a conjecture.