• Michael
    14k
    @frank introduced me to the Collatz conjecture yesterday and I found it very interesting. I'm not a mathematician by any means but I thought it would be fun to look into it, just as an exercise. I'd appreciate feedback from any mathematicians as to whether I'm taking the right approach.

    This isn't an attempt at a proof. Rather it's an attempt to reformulate the conjecture.

    I started at the end and worked backwards through the logic. If the conjecture is true then for all positive integers , and where are some natural number, one of these equations is true:



    The first thing I noted is that the first equation is equivalent to the second equation where , that the third equation is equivalent to the fourth equation where , and so on down the list, and as such are superfluous and can be removed from consideration (these removed lines apply only to odd-numbers, the remaining lines to both odd- and even-numbers).

    I then solved each remaining equation for :



    Given that I decided to simplify such occurrences to . I also decided to multiply both sides of the equation to remove the fractions:



    The equations show a strict pattern which I decided to make more explicit:



    If the conjecture is true then for each positive integer one of those equations is satisfied.

    At this point I am confident in my reasoning. However, what I would like to do is express the above in a more succint manner. It is here that I am less certain. I believe that I can do so with sigma notation, and so this is my attempt to reformulate the Collatz conjecture:



    So, is my notation correct? And if so is it a useful step in trying to prove (or disprove) the conjecture? Or has this been a complete waste of time?

    (Side question: is there a better way to symbolize variable powers of 2 than ?)
  • Deus
    320
    The interesting bits I took from the video is what happened with negative numbers.

    Maybe Toa got close.

    Brute force as was stated cannot disprove to give counter examples. And as was properly stated in the video a different approach is required to disprove it.

    What that is remains in the realms of pure mathematics.

    My two cents on it is the inclusion if of taos research and proximity to solving the problem with a synthesis of what happens with negative integers
  • Deus
    320
    Also to add your invokation of Pi into the equation is also interesting and could be a fruitful area of research in your attempt for proof/disproof.
  • Srap Tasmaner
    4.6k
    So, is my notation correct?Michael

    Unless I'm missing something no, because it's the first term plus the sum of a bunch of products, so you need a sigma not a pi.

    I know the Collatz conjecture, but not much about attempts to prove it or disprove it. (Was it Halmos who said our mathematics is not ready for the Collatz conjecture?) We can assume it can't be proven directly by mathematical induction, or it would have been done long ago. (And it's probably obvious why it can't be, but I would have to think about it for it to be obvious.)

    So what do you do when you can't use mathematical induction? It was a long way around to proving Fermat's last theorem, through more general results in algebraic geometry that included Fermat as a special case, IIRC. Yikes.
  • Michael
    14k
    Unless I'm missing something no, because it's the first term plus the sum of a bunch of products, so you need a sigma not a pi.Srap Tasmaner

    Of course. Thanks for correcting me. Fixed in OP.
  • Deus
    320
    Unless I'm missing something no, because it's the first term plus the sum of a bunch of products, so you need a sigma not aSrap Tasmaner

    By no means a mathematician here either but what truly prohibits the use of Pi in the above equation.

    Happy to be informed.
  • jgill
    3.5k
    Was it Halmos who said our mathematics is not ready for the Collatz conjecture?Srap Tasmaner

    Paul Erdős

    Did you have this sort of thing? (I use that all the time)

    Have fun. It looks dreadfully unappealing.
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal
youtube
tweet
Add a Comment

Welcome to The Philosophy Forum!

Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.