Collatz Binary Decision Tree (Much Improved Version)

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Collatz Binary Decision Tree (Much Improved Version)

Postby frog42 » Thu May 08, 2014 4:07 am UTC

So I realize my last post was likely very difficult to read. I've tried to improve the clarity as much as possible. I'm also including a recursive decision tree for illustrative purposes.


The binary notation of any number to be tested in the Collatz Conjecture contains the complete list of instructions to be followed for this decision tree's output to reduce to 1. That is not to say that a binary "1" always means "Yes" or "No", just that, given the preceding digits and irregardless of the trailing digits, it will always maintain the same meaning.

The decimal number "5" is "101" in binary. However, it takes 5 steps through the linked decision tree to reduce to 1. It can be understood that an infinite number of zeros may be appended to a binary number in order to extend its length without changing its value. In my case, I read binary "backwards" or "mirrored", so I'll append the zeros to the right side. Thus 101 == 10100. Since the decision tree first looks at the smallest binary placeholder, I'm sure you can see it's more convenient using this "mirrored/backwards" binary notation in this situation, allowing us to read left-to-right, digit-by-digit, decision-by-decision.

You will note that any number that starts with 10100 will follow the exact same path down the decision tree. That is to say, 101001, 1010011, 1010001, etc. all share the same first 5 decisions.

Since I haven't seen this information anywhere else and it seems like a rather significant part of the problem, I am looking to confirm whether it is currently an understood factor of the Collatz Conjecture.

Some examples:

[0] 00000 = YYYYY
[1] 10000 = NYNYN
[2] 01000 = YNYNY
[3] 11000 = NNYYY
[4] 00100 = YYNYN
[5] 10100 = NYYYN

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Re: Collatz Binary Decision Tree (Much Improved Version)

Postby z4lis » Thu May 08, 2014 5:23 pm UTC

First, you should probably just edit the original post, rather than reposting and then... rethreading?

Second, when it comes to mathematical discoveries, there's a ton of stuff that mathematicians "know" but have never actually been bothered to write down, mostly because if you know the techniques they know, the problems are sort of evident to them. That's not to say that those things aren't worth knowing or discovering on your own!

Anyway, looking at your example with the verbose path... you say that anything that starts with 10100 for its binary representation will turn in to something you divide by 2 four times. Well, what that means is that the number can be written as n = 5 + 2^5 c for some c. This is certainly odd, so we multiply by 3 and add one, to get 16 + 3 * 2^5 c, and it's evident that we can perform at least 4 divisions by two to this thing to arrive at 1 + 3 * 2 c, which is odd. But now what? c could be anything. The observation you've made is that "if I know the beginning of a number in binary, I know what the first few moves in the Collatz procedure are". This is an important observation, but not terribly surprising. Questions I would think about from here are: If I know the first N digits of the binary string, how many Collatz moves can I know? What if the binary string is of a particular form? But you need precise statements. I also highly doubt anything about orbital mechanics will help you understand the Collatz conjecture, other than the fact that you need math to work with both.

I also don't quite understand the point of the wave path, verbose path, etc... if you just want to write down the moves you make while simplifying a number before hitting the 1-2-4 sequence, then something like 5 = [,4], 16 = [4], 3 = [,1,4] pretty much compactly summarizes everything you'd want to know. Probably nobody is going to want to figure out what you're talking about with the wave path unless you have something useful you can do with it. Blindly transforming data until something cool pops out is fun, but until something cool pops out nobody is going to care to follow what you've done unless they're really interested in whatever you're talking about.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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