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The Collatz problem

by Bernhard Hanreich

Translation by ANGUS MILNE


What is the Collatz Problem?

The Collatz conjecture/sequence arises when one takes any number, x and proceeds as follows:

If the number is even, halve it.

2x => 2x/2                        x is a natural number

If the number is odd, multiply it by 3 and add 1

(2x-1) => (2x-1)*3+1            x is a natural number

Continue this process as demonstrated until…?

Collatz maintained that this process would always result in the sequence: 4, 2, 1, 4…

The conjecture has been applied to many natural numbers, up to and including astronomically high numbers. All these numbers result in 4, 2, 1, 4…

This is most beautiful and has been demonstrated time and time again. However, this empirical evidence does not constitute a mathematical proof.

There are still further unanswered questions, which I can now understand:

1)    Could a loop arise, in which the process begins to form a circle and does not end with 4, 2, 1, 4?

2)    Is there a scenario in which the process could continue infinitely?

3)    Could a number be left out?

Furthermore, mathematics would like a formula by which one can take a number, x, and then know exactly how many steps it will require until:  firstly x is reduced; secondly x results in 4, 2, 1, 4 and thirdly x reaches its highest point and where this highest point lies.

  1. Answer to question 1: No.
    Answer to question 2: Only for an integer x where ƒ(x)=x*2^∞
    Answer to question 3: Does Collatz Conjecture (Collatz-Lewis Theorem) provide for any omissions?

    For data to support these claims, see:

    • Thank you for your comment
      Answer 1: That is what you can see in my graphs, too.
      Answer 2: f(x)=x*2^∞ is ∞ so that is no proof but the rules I found do not allow it to continue.
      Answer 3: the collatz provides for any omission! But that is my opinion and I am sure about it. If you look at my graphs you will see. But it is not accepted till now as a proof.

  2. As to answer 2, ∞ is a concept of method. One only need pursue any train of thought sufficiently to understand that a principle can continue to be extended as required, ad hoc, infinitum. Thus, infinity supports the need of the formulation to continue without necessarily having to continue the formulation.

    For answer 3, allowing x*2^n where n=0 and x is every odd integer, x*2^n where n=>1 necessarily includes every even integer. Every 3(x*2^0)+1 can be mapped to its specific x*2^n where n=>1. No omissions are necessary.

    Thank you for the consideration of my comments.

    • 2 : What is ∞ As you said it is a concept an Idea You do not have to proof every number, if you understood the principles and rules in the collatz for example. The by the rules repeating symmetries are proof enough to accept it, ad hoc, infinitum. Is this what you ment? Then you would agree that the rules I found are a kind of proof.
      I am not native english speaking, so it is not so easy to understand all refinements. Please tell me if I misunderstood.
      By reading your comment I remember a thought i had .
      Is there a smaller ∞ and a bigger ∞ like two circles with different size. Is size real or only a question of proportion or dimensions or point of view. If you pursue this thought in the collatz you will find that the road to return from ∞ to 4,2,1 is endless too. That is the reason for me to say that there are different ∞ qualities. ending ∞ on one side or even ending on two sides or without there an end at all or smaller and bigger, longer and shorter ones,etc. It is funny to realize, that the distance between 0 and 1 is endless, too, for example.

  3. I follow the Aristotilean understanding of ∞ being only a potential to keep going.
    One could write the number line as 1, 2, 3, … implying ∞.
    One could also write the number line as
    1*2^0, 1*2^1, 3*2^0, 1*2^2, 5*2^0, 3*2^1 . . . also implying ∞
    With either version, ∞ implies you may continue, but if you choose to stop, whereever you stop is finite.

    I sent you a note because you were the most recently active posting on another forum I found with regard to this matter. I thought you might be interested in seeing an approach that I have not found anywhere else on the internet. Your collection of graphs only strengthens my conviction of what I also have discovered.

    • Thank you for being so kind to answer and to give me the links.
      Yes ∞ keeps on going. That is the same I think. But still there are some Borders possible. You can take for example 0-1 and divide it 1/2^n and you get an endless line between 0 and 1 => ∞ with 2 borders
      You can take the natural number N. 0=>∞ 0 is a border but Z does not have a border but a center ∞∞ that N never will have. So the quality of ∞ is different, also it means the same keep on going as you said.

      Thank you for telling me that my graph had strengthened your conviction. It makes my happy to here that because that was the reason for me posting them in the net. It shows me that it was good to do that step.

      I looked at the link, I like it. Thank you. I will sone answer there. But as fare as I could see till now there is nothing that you cannot see in my graphs written there. But I must say I did not have time to read it carefully.
      The endless and detailed way of growing and the growing symmetry, that you can find in my graphs, I could not find anywhere else till now, too. Specially these two phenomenons but not only these two do proof the problem from my point of view. But I am very curious to hear why not, if you can tell.

      • I have to add that the Symmetry you can see is an ENDLESS ∞ growing symmetry. That is the reason I was occupied with the ∞ phenomenon.
        “One only need pursue any train of thought sufficiently to understand that a principle can continue to be extended as required, ad hoc, infinitum. Thus, infinity supports the need of the formulation to continue without necessarily having to continue the formulation”. As you said.

  4. Your illustrations concretized the value of preparing one to aide with the clarity of how the formulas relate to the data. It has been prepared and is now attached. It should make it easier to decipher what is going on without trying to go though all the thought processes the thread contains leading to the conclusion. Thanks again for some additional thoughts on the matter.

    • You are wellcome.
      From my point of view the graves not only made it easier to understand what is going on, it solved the problem, because you can see the rules and from that you know that it has to be true and that was the Question. Is it true or not. As I said already,from now on it is only a question of translation( you call that dechípher). I am sure about that collatz is no conjecture any more. It´s a totally logic result from the graphs to realize that Collatz has to be true. I know that this is a way to proof that is hard to accept because it is no simply calculated proof but it is possible to see if one wants to.

  5. Yes, that is the bottom line. Does it meet the criteria of being true. If so, it is. If not, it remains a state of yet being indetermed.

  6. Indetermed should have been indetermined.

  7. So if something is not understandable or unclear (that is how i understand indetermined. I hope that is right) give me the chance to answer your question without speaking around like a spiral getting closer and closer to the point but not being able to reach it. As you said( I use other words) I found the connecting part between the formula and the data. This makes it clear how the data fit in the process of the formula.
    It also shows as you can see that these processes are endless so no more calculation is necessary to proof the increasing part nore the reducing part.
    That it is still a long way to calculate from a number down to 4,2,1 is a question of finding the reducionline. As i said this is an growing process, like life. Look at a tree and the process of growing. it is clear and easy for a tree to grow step by step. Finding the reason why the materials, which the tree needs to grow, come and where they came from is not that easy any more. Do you need a proof that the tree exists, when you are in front of a tree. You will hit your had if you do not accept it and try to walk through.

    My graphs are a oneway proof, as I call it.
    That is no unusual thing in mathematics starting with the easiest calculation at all.
    1+1=2 is an axiom and it is true for accepting it like that.
    but what about 2=1+1 that is not that easy anymore.
    because 2 has endless other possibilities
    So in one direction it is the only way to be. 1+1 = 2 it can not be any other number
    in the other direction it is just one simple way how it could be
    If you know where it came from it is the only possible result but if you ask what is two? 2=?
    the result is???? You will never be able to answer totally, or at least not that easy. You have to describe it like every calculation of numbers that have the result with the size of 1+1. Or like that in better english.
    And this is the same with all the other numbers. Do you ask to find the way back if you do not know where they come from.
    Still an open question??
    So that is the same with the Collatz. If you know the tree below a number it is easy the find the way. If you do not – have a nice time! Look at my graphs you will see what I mean. You need to know the growing symmetry!!!! It was the growing symmetrie to be found to proof the Collatz.(that is how I see it)
    I was given this that is what I know but do not ask me where from. I would have to tell you all about my life and then still you would simply have to accept it

    If there is an other way to see it or still a question. Tell me

  8. Validation is a process every individual has to perform for themselves. All we can do is point to the evidence. It is up to them to assimulate or process the evidence for themselves.

    I had to make some corrections to the first sheet. I combined it and the next one with one additional chart. I only hope the formulas and methodology laid out are clear enough that those who seek to understand it, can and may..

    • Who is we and who is them. Where did you point at? Which corrections did you have to make? Is there a mistake in my thoughts or something missing?
      How should understand it? For me it looks clear but I did it and know it quiet well but everybody can ask if he does not understand what I mean. I´am a little confused

  9. We, would be those who beleive it to be true, and more or less demonstrated.
    Them would be those who don’t I guess.

    We point to the evidence, or the demonstrations,, graphs and other materials.
    If a mistake can be found in the math or the method of approach, then the evidence is incomplete.
    Then it comes down to “What would constitutes definitive proof”?

    The corrections I made where in the top lines describing how to solve the columns working down.

    For me, I understood it when I saw how it worked. The inter-relationships of the formulas described what made it work.

    As to axioms in math, I understand they differ from say an axiom off philosophy, but my math skills were learned 35+ years ago, and only to an introductory of trigonometry.

    When I’ve looked at some of the proofs that were submitted and rejected by the “gatekeepers”, it’s like looking at a foriegn language. I would need a translation dictionary to understand what the proof is proposing, and why it was rejected.

    • It made me so happy to hear that you could understand what I mean that I got tears of happyness in my eyes. For me it was a work I did after a cancer operation to forget my sadness about what had happend to me. It makes me so happy to realize that I still can do something and to be understood. Thank you so much!!!

  10. You’re most certainly welcome.

  11. Greg Lewis permalink

    Just cruising some of the links I bookmarked a while back. If you’re interested in another development on this, I created this presentation:

    • Sorry for not answering earlier. I have a hard job. It looks good what you did. Do not understand jet, but i will try to. Thanks a lot for the link! Very interesting.

      • Greg Lewis permalink

        If you have a specific question on it, I can try to answer. As your statement is posited, I can only struggle to guess at where any confusion might lay.

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