Jump to content
Objectivism Online Forum

Just_a_hobbyist

Newbies
  • Posts

    7
  • Joined

  • Last visited

Previous Fields

  • Relationship status
    No Answer
  • State (US/Canadian)
    Not Specified
  • Country
    Not Specified
  • Copyright
    Copyrighted

Just_a_hobbyist's Achievements

Novice

Novice (2/7)

0

Reputation

  1. Yeah, I started with the first few thousand integers, halved until all everything that could be halved had been, and then ran the algorithm for a fixed 400 passes, to see whether the value at the 400th pass was a 1,2, or 4. Exploring the first value revealed the 6n+4 form, from this I tried to go down the 3n+2, 3n+1 root etc, but realised you get into crappy decimals, so looked into ways of conversions. My early attempts are contained in my first real post I made here I used 2X which was ultimately 6n+1 and 6n+5 combined. I can't say I've tried to relate it to Fibonacci, but with how I've broken it down now, with n values and the 6 forms I'm left with seems (12 if odd and even n are treated differently) seems like the solid foundation to work from. I have the opposite view with regards to that talk, but mainly because he said that he had found what I have found so far. Since I'm not a mathematician, and he's been working on this for years, I use it as validation, that perhaps my prime nodes do offer something new. No one had reported the prime sequence I use to generate my nodes, the sequence of numbers was ungoogleable beyound 157 as a coma separated series. The key ratio to this is 1.5, and he states about it going to approximately 1.5*googol for example. The talk to me was meaningful, but then it would be in my interest for him to say these things, so I guess that can also be taken with a grain of salt, lol. It is true though in the sense that even if you start with the highest known odd integer, a higher integer will be encountered, and because of this there is nothing to say it cannot go odd-->even -->odd-->even infinitely, so a proof under the terms of Collatz using a 1-->1 approach I.e that 11-->34 and only 34, cannot work infinitely. or at least this is my understanding, actual research articles on this subject are out of my reading ability so everything I've done is a self exploration.
  2. This is essentially what I realised. I broke the problem down into 6 forms which covers every integer exactly once. But I discovered that whether n was even or odd within these forms dictates the real behaviour http://i.imgur.com/G3ifygC.png one pass interactions (this pattern goes on infinitely) http://i.imgur.com/TSHi0g3.png The grand scheme of things: (the number of paths a node has cannot exceed 8, these nodes are based on an infinite sequence of primes I discovered looking at Goldbach) My work is also contained here. http://mymathforum.com/number-theory/48561-twin-primes-goldbach-collatz-2.html I can only say I believe this would underpin a proof, but am as yet unable to produce a formal mathematical proof.
  3. I didn't mean to disappear, I couldn't figure out how you had derived that sequence of integers, and then I started looking at Collatz from an entirely different perspective. My work is here http://musingsofminers.wordpress.com/ Have you two made any progress?
  4. I began by exploring if there was a bias to which "phase" numbers were in when they terminated, if 400 steps of 3n*1 or 2n are applied regardless of whether it took 3 steps or 303 steps to reach 1, the number at the end of the 400th cycle would be either 1,2 or 4. while I observed a visual pattern in this, there was no overall bias to the "phase". Next I explored how consecutive numbers were in phase how many pairs of numbers both were in phase of the 4,2,1 cycle how many triplets, qudruplets etc. Up to strings of 32 consecutive numbers being in phase. Again I saw a general pattern but nothing mathematically defineable. http://imgur.com/q3DVOWa You can clearly observe what i explored as consecutive integers being in phase, it is where the colours lined up I then explored whether if at the 400th iteration would pairs more likely occur if they ended in 4,2,1. but again there is no bias, 1 appears to form longer pairs, but that may be due to sample size. Next I explored if there was a "prime bias" as to if I started with a prime number, would it be 4,2,1 at the 400th iteration. But again there was no clear bias, the primes appear to terminate evenly across 4,2,1 On being surprised that what I expected to show bias didn't I began to investigate if the numbers had "forms" (this relates to previous work on Goldbach) I realised I could ensure that every number was in "phase" if I started with N=1 --> infinity, if odd --> pause, if even divide repeatedly until odd was reached. There was then 100% certainty my current number was odd. Then I applied the algorithm. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (here is my initial explanation which I wrote when I first went to reply to your thread, I have re-evaluated it to the 2nd version shown outside of these lines) Final edit: added this: http://i.imgur.com/34kvzE2.png This is my actual first draft, I used 2X before I realised it was of the 12n+2 -->6n+1 odd Your number can only be odd (N) Multiply that number by 3 and add 1 3*(N)+1 Your number can only be even it has the form 6n+4 since 6n+4 is even, it can be halved to 3n+2 3n+2 3n+2 is either odd or even, it has 3 potential forms 3n+2 = odd -->3*(3n+2)+1 --> 9n+7 (has form 6A+4)-->3A+2 3n+2 = even (it is of the form 12X+8)-->6B+4 -->3B+2 3n+2 = even (it is of the form 12X+2)-->6C+1 -->18C+4 (which has form 6D+4) --> 3D+2 Here are some inequalities: 6n+4 = 3*(N)+1 > (N) 6n+4 > 3n+2 > (N) 6A+4 = 9n+7 > 6n+4 > 3A+2 = 4.5n+3.5 > 3n+2 > (N) 6n+4 > 3n+2=12X+8 > (N) > 6B+4 > 3B+2 6n+4 > 6D+4 > 3n+2=12X+2 > 3D+2 > (N) > 6C+1 -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- My current explanation: (about 8 hours later) Since N was odd, and 3*N+1 was even, a smaller set of even numbers exist which would be made directly from this initial step from odd. They are all of the "6n+4 form" which obviously is halved to "3n+2" and that is where the fun begins. But the value at "3n+2" has three distinct forms it is either odd of form "6W+5", of form "12X+8", or of form "12Y+2" I.e. it is either odd (which has form 6W+5), so multiplied by 3 and 1 is added, transiently 18W+16 --> returning to the "6n+4" form. but with a different variable ("6A+4") it is of the form 12Y+2 it can be halved only once to 6B+1 at which point it returns to the "6n+4 form" but is equal transiently to 18B+4 (also 6C+4) it is of the form 12X+8, it can be halved to 6X+4, and halved again to 3X+2 where there are multiple paths it can take I realised the route it took depended on the starting odd number, which can be split into 3 forms: 8n+1-->(12Y+2) 4n+3-->(6W+5) 8n+5 -->(12X+8) if we use a small n, which can take values 0 to infinity and use it to construct the numbers: (out of the first say 5000 numbers) Total times 1250 8n+1-->24n+4 --> 12n+2--> 6n+1 A reduction of 2n (-2n) 2500 4n+3-->12n+10--> 6n+5 An increase by 2n+2 2*(2n+2) 1250 8n+5-->24n+16--> 12n+8--> 6n+4-->3n+2 A reduction of 5n+3 (-5n-3) This leads to a net reduction. It is of "-3n+1" The key principle is redefine each number at an appropriate step into ta different form. This prevents spiraling into decimal values of n + decimal values and huge values because the 6n+5 form increases, which I believe cannot be shown to decrease as it can always potentially create a larger odd number. Ultimately when the 4,2,1 cycle is reached, it comes from 6*variable +4 --> 12*variable+2 --> 6*variable +1 where the variable is zero, at this point the variable cannot be converted into other forms, and this is why the 4,2,1 loop occurs, cannot be escaped, and why every integer will enter the entire Collatz cycle and eventually reach 1, before repeatedly cycling to 4,2,1. (this is my hypothesis anyway) I believe this is where your tiers come in, you notice the + 32's, 64's etc powers of two. e factor, I look at it through the fundanental value of n which makes up the number. (I admittedly haven't spent enough time going over your work compared to my own, but this is how I perceive what you have presented. It describes the comb like graphs which can be generated at least) Returning to my work: 8n+1 decreases by 2n (has n value of 4n and then n=n) (0,4,8,12...)-->(0,1,2,3....) 4n+3 increases by 2n+2 (has n value of 2n+1 and then only 3n+2) (1,3,5,7...)-->(2,5,8,11...) (1-->4-->2, 3-->10-->5,5-->16-->8,7-->22-->11) 8n+5 decreases by 5n+3 (has n values of 4n+2 --> n=n) (2,6,10,14...)-->(0,1,2,3,4...) example from 8n+1 SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg from 4n+3 SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg (I believe it is only needed to go to the ODD, but I continued this for some reason) it demonstrates that the "n" values follow the collatz conjecture I think? because N started off as odd, in converting a form, it causes the "n" values to be initiated into the Collatz cycle, just like we did by initially ensuring only odd values could begin the cycle. Because only n values here are odd, when manipulated, they enter the cycle. (This isn't coherent really, I'll return to it at some point) from 8n+5 SEE IMAGE THE TABLE BROKE http://imgur.com/k7PQWTg I think I've shown the Collatz is contained within the Collatz? (again I'll return to this, my work is as I say in it's infancy atm) http://imgur.com/0EQymYv The values in green are unique values, it shows that odd numbers enter via 8n+1, 4n+3, 8n+5 but can only exit via being even in the form 8n+5, they either at this point halve further to 1, or halve to an odd number which previously cycles to 5,7, etc. a number which can be written as 3n: 3,9,15,21 cannot re enter the cycle unless it is the result of a halved even number. But this has already occurred previously, because all halvable evens were halved to odd before the algorithm was applied. (also starting to lose clarity, I'll readdress this at some point) http://imgur.com/3UmbjRz This is the image I had obscured, I guess it didn't really detail to much, just demonstrated what I'd use as forms, of course with this information my work is repeatable but they've already been explained above. http://i.imgur.com/ZZ6B1zl I generated this exploring average values (I won't detail the method, but this is the result of it across 400 passes as mentioned earlier) http://i.imgur.com/08mZo5N has similar origin to the graph above, this is looking at the tail end of what occured in the first graph that seemed to reach almost zero. I have an array of beautiful graphs, I have not seen replicated through an image search, but they don't really add too much of value, except it is evident that there is a cycle going on here and with each stage something is filtered out, leaving yet another unique graphical pattern (I like graphs, I guess that is also why I noticed a fair amount in a short time, as I just generate mass amounts of data and graph it, if there is a pattern I analyse it, if not I try something else) This is quite a large part of my work, but not all of it, and I'm still looking to explore deeper. I would be interested your thoughts on how you perceive what I've presented relates to your "tiers", and perhaps if you have any questions it can guide my direction to further explore this. I suspect not much of this is novel to you, but I just wanted to demonstrate I really did formulate this independently of your work, although I've since seen the discussion here and on mymath. Which I guess means I can't claim anything I do from now i did wholely without influence. But I was just surprised that by Googling some of my findings I found this particular thread, and you had only just yourself made a video of your findings. Good luck to you, and maybe we can collaborate on this? this is what I'm willing to release publicly at the moment, Based on your statement that your only protection to your work is what you have time stamped through usinIg this, and my Math. I will do the same, which i why I used an initial reference number at the start of the message, It is so I can prove when I released my findings. Sorry if I came across negatively, but I'm sure you can understand my position. ------------------------------------------ For my reference: 1928372
  5. As a final point so to make an actual contribution: this stopping time is the result of the fact that some numbers reach a certain point for instance 22 is reached once from below (7) and once from above (44) But the number 21 is not in this position, there is no number which when multiplied by 3 and 1 is added makes 21. 24 is also not in this position, because there is no number which when multiplied by 3 and 1 is added makes 24. 19 could hypothetically be in this position it is half of 38, and 6*3+1 = 19 EXCEPT 6 is halved to 3, rather than multiplied by 3 and 1 added This has something to do with what I referred to as "forms" which is similar to Dream_Weaver's "tiers", but the exact details I have not yet disclosed, but it is an interesting observation, at least it has a role in my work on the subject.
  6. I was unable to edit my post, but please let me attach this onto the bottom as I may have come across the wrong way: Edit: firstly an explanation: I decided to post this after trying to write various messages here and felt the need to say something, which wasn't disclosing my entire work. I admittedly didn't look at the video presented closely enough, I was more shocked that within days of me observing something someone has posted something similar to myself. I have since looked again multiple times and can see why we have used different variables. You have produced quite an elegant explanation of some of the graphs I have generated I can only presume you have seen the same images your self. http://imgur.com/ZzwQp2u This is my original work, the graphs may look familiar / the method I used to break down the numbers into "forms" is essentially the same process as your "tiers" but I have used different variables afterwards (I am aware the image is too small to resolve my work except from the distinctive pattern of the graphs, I have done this for 2 reasons: To show there is some truth in what I say, and also to protect my work initially, but believe me, I think we can both help each other here) Like I say, I very much want to prove this conjecture, and I come seeking potential collaboration, but I want to safe guard my work also. If nothing else, take hope from the thought that someone else has independently arrived at the same method as you, as I have knowing I'm not crazy and someone has stumbled onto the same thing I have when exploring numbers. --------------------------------------------------------------- For my reference: 6791028
  7. Hello, this is my first post but I found your work through Googling what I discovered personally and independently, it seems we are on the same path, my work is in it's infancy however as I've spent a week so far on this, (I have spent the majority of time previously on the Goldbach conjecture and found a possible similarity hence applied that to Collatz and here I am) The reason I'm so far in a short time is: I started initially at the point where you have reduced everything to Odd's, I used the exact same starting point, which is why I thought what I'd come across was novel as the resulting sequences from that do not appear in oeis.org, while the http://oeis.org/A000265 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11... has been used in other areas of maths. But 4,4,10,4,16,10,22,4,28,16,34... is however a novel sequence (except you have already encountered this in your work) But I believe I have what you have in terms of tiers, except I have a general statement of that specific observation. The variables I have used are different to your's but generate the same pattern. If you would be interested in a collaboration on this I will happily share my work but would rather do so privately for the time being. --------------------------------------------------------------- For my reference: 6791028
×
×
  • Create New...