Chapter 3.4:
Will There Be Patterns?
In the early days of my research, this question fascinated me. Will there or will there not be the familiar patterns in the Stopping Times and Max Value Graphs for non-integer worlds? I so badly wanted to know! I invented my own technique for defining the xa+b Collatz World during the summer of 2018, yet I had to wait all the way through the fall as I learned to program Python to see the results.
Playing around just with paper and pencil, however, I did acquire a decent body of evidence that (probably?) the non-integer worlds had the familiar mysterious Stopping Times structures. I had found a few examples of what appeared to be Stopping Times Bands. Recall that the Stopping Times Bands are cases where many input n values near to one another seem to (coincidentally?) inexplicably share the same stopping time, aka they take the same length of time to fall to a loop. Often they do this even whilst taking wildly different paths to reach the loop. It's interesting.
Let's take a look at an example of this within the x3.5+1 world:
1431 >> 5008 >> 2504 >> 1252 >> 626 >> 313 >> 1096 >> 548 >> 274 >> 137 >> 480 >>
>> 240 >> 120 >> 60 >> 30 >> 15 >> 52 >> 26 >> 13 >> 46 >> 23 >> 80 >> 40 >>
>> 20 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
1432 >> 716 >> 358 >> 179 >> 626 >> 313 >> 1096 >> 548 >> 274 >> 137 >> 480 >>
>> 240 >> 120 >> 60 >> 30 >> 15 >> 52 >> 26 >> 13 >> 46 >> 23 >> 80 >> 40 >>
>> 20 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
1433 >> 5016 >> 2508 >> 1254 >> 627 >> 2194 >> 1097 >> 3840 >> 1920 >> 960 >> 480 >>
>> 240 >> 120 >> 60 >> 30 >> 15 >> 52 >> 26 >> 13 >> 46 >> 23 >> 80 >> 40 >>
>> 20 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
1434 >> 717 >> 2510 >> 1255 >> 4392 >> 2196 >> 1098 >> 549 >> 1922 >> 961 >> 3364 >>
>> 1682 >> 841 >> 2944 >> 1472 >> 736 >> 368 >> 184 >> 92 >> 46 >> 23 >> 80 >> 40 >>
>> 20 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
1435 >> 5022 >> 2511 >> 8788 >> 4394 >> 2197 >> 7690 >> 3845 >> 13458 >> 6729 >> 23552 >>
>> 11776 >> 5888 >> 2944 >> 1472 >> 736 >> 368 >> 184 >> 92 >> 46 >> 23 >> 80 >> 40 >>
>> 20 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
So what we have here are five successive input n values each of which takes exactly 33 steps to reach 1. This is so cool! We definitely seem to be looking at a small piece of a horizontal band on the Stopping Times Graph here.
Just to make sure we're clear on one fact: The horizontal bands are definitely not always made of successive integers. They're a little random. In this case, the numbers (near each other) which take 33 steps to fall are precisely: 1420, 1421, 1422, 1424, 1426, 1427, 1431, 1432, 1433, 1434, 1435 and no others. Note the strange sort of bizarre "line" that these form.
I had enough data to make a hypothesis. There seem to be x3+1-like patterns in the x3.5+1 Collatz World's stopping times. Now to test that theory out. I spent the fall learning to program Python. Over the summer of 2018 I finally got to "press go" and see what these non-integer Collatz Worlds looked like!
I could not have been more excited.