Working on prediction of Collatz trajectories rather than on solving the Collatz conjecture, I found some interesting properties.
This is related to the Collatz matrices. See link below.
I realised that I can get the exact same number in the column C1 (C1) if I do 2k-1. Example, from k = 3, I get a 1 on top of the 3. 3x1-1 = 5, I get the same 1. 5x2-1=9, 9x-1 = 17, etc. Going that way, I can get the exact same 1 on top of a large number.
k=3, 1 in C1 and in C2, 1 in C3
k=232769, 1 in C1 and 49153 in C2. Observe that the other numbers are large, but the 1 repeats
Of course, this also corresponds with the presence of large powers of 2 as divisors. Same for k = 5, there is a 3 on top of the 5.
k = 12289, 3 in C1 and 18433 in C2
I am only posting 2 examples, but that works for any k. I also discovered that, by doine 4k-1, I can predict the number in C2. As you can see from the previous examples, that number is not the same in each pair of matrices.
For k = 3, 4x3-1 = 11.
k = 11, 1 in C2, same as in K=3, and 49 in C3
By doing 64k-21, I can predict the number in C3 when k = 11. 11x64-7 = 697
Of course, this is not magic. The equations predict that.
The numbers in the colored row and part of the Collatz sequences. One of the link about the matrices:
https://www.reddit.com/r/Collatz/comments/1s1d02t/creating_collatz_matrices_using_a_spreadsheet/
More links in my profile