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Collatz Conjecture might have been solved

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Well friend, I, like many, spent countless man hours working the Collatz Conjecture looking for some insight.

Maybe there is something to help us crack the first form the 3X+1 form in the 3(X+1) form.

 

As for my conjecture that all multipliers are seen in a cycle well I think the multiplier 9 is an example of a divergent system. Something we have been on the lookout for with the 3X+1 form

 

So I'm happy to point this second form out if I am actually pointing this second form out for the first time and welcome any prior art. I'd love to read about what was learnd from the second form if there is anything on it already. 

 

 My current project is a new type of sort for data compression so I am not planning to work this second form any time soon however, I do have a site eberg.us and I started a thread there so if you don't see me here friends can post there if there is something magical to this second form.  It interests many I am sure.

 

 

Ernst

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If nothing else, you reminded me of the man hours I spent analyzing the Collatz. Over the weekend, I submitted an inquiry to an agency that provides tutoring to inquire about finding someone that can assist me in converting my observations into a form that I might be able to submit for recognition. Godspeed on your latest project.

Edited by dream_weaver

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That sounds great. Be sure to send me a copy or link.

 

I was thinking that nature doesn't like "one-zees" and now we now have two examples of the Collatz type of cycle structure in two different algorithms.

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EC    16

Besides proving something that is likely true but previously unproven what would be the usage of such a proof except for being mathematically rigorous? In other words does proving this tie into anything else that's known to be useful in anything outside of abstract mathematics?

 

Or leave out the last question because I'm also wondering if any other abstract math problems hinge on this being proven true or if it's just a stand alone thing that ties into nothing else known so far?

Edited by EC

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One benefit to myself has been getting a better handle on understanding just what constitutes evidence and how to present it correctly.

 

Robert Knapp, in his book "Mathematics Is About The World" is applying Rand's framework using Euclid's proofs and the Axiom of Archimedes to more fully understand how mathematical concepts tie to reality. So to address the question of what use would such a proof serve would be premature at this point.

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If I may, I love the Cycle. I have been attracted to all things that cycle. I spent many years on the Collatz Conjecture. I studied it.

 

for Collatz it's in a class of [A(X),+/-Y,X/2]  Where Y is odd and X is usually 3

for this new form [A(X+/-Y),X/2] where the A scales for A > 0

 

when A = 1 the cycle is { 2 1 } cycle notation doesn't use commas

when A = 2 the cycle is { 4 2 1 }

 

So the cycle structure is there. independent of the two forms. Cycle may well be a force in Nature.

Elementary particles spin and so on.

 

If you guys like you can check out http://arxiv.org/abs/1405.2846 Introduction to Dynamic Unary and see a lot of cycle that scale from 1 to infinity. If you have questions you can contact me at [email protected] the website is eberg.us

 

I worked with [[A(x+1),x/2] yesterday and I feel it is as stable as Collatz. True it is not Collatz in it functioning but it works well in a 1 to 1 bit ratio for input number size and output number size when the actions odd = 1 and even = 0 are captured as a file I have experimented with Collatz to make data transforms and this form ( Berg? ) seems mathematically true as well..

 

I am still searching for Prior art on [A(x+/-Y),X/2] I mean I would love to say I saw it first but have I?

 

In conclusion the study of these dynamic equations is the study of Cycle which may well be a force of Nature.

 

Thanks for the ramble.

 

Ernst

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And yet another pattern emerges, but it will wait until I decide which tutor(s) to interview for the opportunity to express it one final time.

 

Ernst, I've only spent a little over a year studying the Collatz. I'm curious what you are calling 'cycle'. I think of cycle, day/night, moon phases, and seasons come to mind.

Cycle carries with it a notion or implication of a loop to me, which the folk over at MyMathForum suggested I had not addressed. As I've organized the materials thus far, I've found permutations that alternate between two formulas which double in complexity depending on the number of iterations are applicable. I show what I mean at the end of point 40 in the post just before this one.

 

As to cycle being a force of nature, math can only serve as the handmaiden of observation for demonstrating that. The observation of a cycle, or pattern can indicate a causal connection. The math enables us to concretize it in a formulized approach.

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I term Cycle in the Collatz and ilk as the part where the integers repeat.

 

In the 3(x)+1 form we have a { 4 2 1 } cycle. in 3(x)+3 we have { 12 6 3 } Collatz conjectured all integers "go to one' I see it as a cycle of  elements and not a "GoTo".

 

So you have been looking at the Collatz for about a Year? LOL it hooks ya.. 

 

Remember there are systems for all + odd  in 3(x)+Odd-Integer. So we are not limited to 3(x)+1, 3(x)+3 and so on.. At some point a cycle of values will be seen.

 We also can have systems of minus the odd value as in [3(x)-1,x/2]

Perhaps we might need to restate : no matter what integer input a cycle of elements will occur.

 

Also if we represent the odd action as a set bit and the divide by even action as 0 then the stream of bits emit from any integer processed to cycle is a parity language that is Fibonacci in structure.

Edited by Ernst

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Hey guys! I was recently introduced to these forums (and this post) and thought I would add an observation I had today.

 

We can say that the number of times it takes for an integer "n" to reach 1 is called the stopping time, or ns. I decided to represent these numbers as combinations of other stopping times + 1. For example, the first 10 would look like this:

 

1s  =  0

2s  =  1   = 1s   + 1

3s  =  7   = 10s + 1

4s  =  2   = 2s   + 1

5s  =  5   = 16s + 1

6s  =  8   = 3s   + 1

7s  =  16 = 22s + 1

8s  =  3   = 4s   + 1

9s  =  19 = 28s + 1

10s=  6   = 5s    + 1

 

What I noticed, is that if I continue writing the numbers out as combinations of other stopping times, each odd integer had a stopping time that paired with another even stopping time:

 

3s   =  7  = 10s + 1

20=  7  = 10s + 1

 

5s   =  5  = 16s + 1

32=  5  = 16s + 1

 

7s   =  16 = 22s + 1

44s =   16 = 22s + 1

 

 

It may also be worth noting that the second number (higher one) increases at a constant rate. So we could write each pair as (n, 6ns+2)

 

This is mostly just an observation, I haven't had too much time to actually see if this can be applied to get anywhere with the conjecture.

Edited by xBlueOblivion

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The ns may have merit when breaking down the permutations in both the home base number to its respective termination point, and the home base number to its ultimate reduction, although the latter has not fully revealed itself in my investigations.You're considering both odd and even in determining the stopping time. Since odd is divisible by two until it becomes odd, it has not been a factor in much of my investigation.

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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

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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

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What I noticed, is that if I continue writing the numbers out as combinations of other stopping times, each odd integer had a stopping time that paired with another even stopping time:

 

3s   =  7  = 10s + 1

20=  7  = 10s + 1

 

5s   =  5  = 16s + 1

32=  5  = 16s + 1

 

7s   =  16 = 22s + 1

44s =   16 = 22s + 1

 

 

It may also be worth noting that the second number (higher one) increases at a constant rate. So we could write each pair as (n, 6ns+2)

 

This is mostly just an observation, I haven't had too much time to actually see if this can be applied to get anywhere with the conjecture.

 

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. 

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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:

The edit functionality here is set to an hour from the initial posting.

 

 

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

I had a look at image. The only part that looked familiar (to my spreadsheets efforts) was the second graphic down on the left. It resonates with the opening slides counting the divisions to use as exponents later on.

 

Some of the other graphics remind me of a few charts I encountered researching what others have tried. I never converted the numbers into charts or graphs for analysis.

 

The "tiers" were found last year. The recent attention gave to this thread gave me pause to re-examine my work, and discovered how to break them down in the systematic approach addressed in the video outline.

 

To the best of my knowledge, there is no cash prize currently offered for this conjecture. So at stake is really only the fame that might be associated of being named the discoverer of the principles that underlie it. The only protection my work on this currently are the date and time stampings of what I have posted thus far. I thought it might be insightful to see the approaches and developments as I discovered them.

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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

Edited by Just_a_hobbyist

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From the tables of numbers I've examined of yours, it reminds me of what I present on slides 1 thru 43 (first 430 seconds of the video, @ 10 seconds per slide.)

 

If you are familiar with the phenomenon described as the "hailstone sequences", slides 44 thru 72 breaks this down in terms of tiers and permutations and frequency of occurrences.

 

The contention is, if you take your 8n+5 set (my baseline), and apply the process illustrated, you will eventually reach a 3n value where n is an odd integer, {3, 9, 15, 21, ...} As you are already aware, (3n+1)/2any power greater than 0 does not produce this set of values. Starting with 27, until you reach 445 - (3n+1) will become odd within one or two divisions.  3(445)+1 turns odd only after 3 divisions. When (3n+1) can only be divided by 2 once before becoming odd, the value climbs, while divisions of two or more times produce the descents.

 

Aside from the video, I've posted just about everything I consider of significance to date in this thread. The advantage I found in doing the video was it showed the step by step in what I perceived as a cleaner fashion than trying to explain it strictly verbally.

 

As to collaboration, I've presented my discoveries to date here. Questions and comments on them to this point have generated further observations. Even in our short exchange, I just found another way to further set up in my spreadsheet the unique sequence of {1, 5, 1, 11, 7, 17, 5, 23, 19, 13, 1 ...}  I have yet to determine if there is any significance to this observation.

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It is good to see that people have an interest still in this.

 

I have an idea for a possible proof and I sure can share such a thing but I would like to work semi-privately with others.

 

My email is [email protected] if you would like to consider an approach that I have not seen done before. Call it an exploratory group email conversation.  

 

 I have enjoyed the Collatz stuff since 1993 and I often include the mechanics in experimental software design so I am not so new to it but will benefit greatly from maths smart friends when it comes to writing things down.

 

Now to go back an reread the newer posts here.  Again great to come here and be a part of such an interesting and honourable quest.

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My contention is that if the 8n-3 odd numbers are reversed to a multiple of 3, all of the odd numbers are consumed in the process. Going forward, all of the 8n-3 values reduce to 1 taking this aspect into consideration. QED.  Putting this into conventional,  formal,  mathematical notation has demonstrated itself to be the difficulty here as well.

 

I've presented this video to www.wyzant.com along with the request for the same as well, but as of yet have had no response. There seems to be a disconnect in the approach to set theory here that may be beyond my capacity to resolve.

 

PS: I've also sent an e-mail to the address you provided.

Edited by dream_weaver

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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?

I presume by 'that sequence of integers' you're referring to {1, 5, 1, 11, 7, 17, 5, 23, 19, 13, 1 ...}. If so, I did not go into it here. If you have access to Excel, it is pretty easy to set up. Some of the Excel compatible spreadsheets may work as well, if it has VLookup functionality built in. It uses a modified Fibonacci approach,

 

Your chart on 'node-relation-to-odd-even-n-public' with the black background, cinched for me that the 6 headings exhausts, includes or, in another word  accounts for all of the counting numbers.

 

As to progress, I've been taking a break from the Collatz lately.

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Well, I had one email about the possible solution to this so I'll just spill the beans here.

 

I believe that we are asking the wrong question when it comes to Collatz Conjecture. By that I mean we needn't prove that integer X ends on integer  y  but let me explain and I welcome your comments.

 

 We try to prove that A goes to B and with little success however what is really happening is that some numbers create a Cycle and the rest do not.

 I believe you all will find that Collatz in the form of [3X+Y,X/2] is just one of an infinite set of Y==ODD.

So 3x+3, 3X+98466894651694615 and so on.

 What we see when we process any integer for any Y is that in the Cycle the Y is seen.

So what we have is a Struggle between 3X+Y and X/2 until an integer that cycles is reached.

I propose the following as the foundation of a Proof.

 

1: That the X/2 part wins over the 3X+Y part.  That this process necessarily reduces all unsigned integers towards zero.

2: That a Cycle of integers occurs when the input to this dynamic system is the Y of the system.

 

So it can be broken down to integers that cycle and the process of the struggle between two functions with one growing the value and the other reducing the value.

 Also noticed is that the even number generated by 3X+Y is the only element that belongs to both the 3X+Y part and the X/2 part.

 

 So, it is not to prove that all numbers go to one it is that the  [Ax+y,x/2] A=3 reduces value towards zero over the long run and that when a value that is in the cycle of the Y is reached an endless cycle occurs.

 

 That is what I have come to believe since being introduced to Collatz Conjecture in 1992.

 

 Now how do we show that the value reduces towards zero? I need help with that.  I think I can describe the cycle action but that is open to share too.

 

 I have written my first scientific paper which introduces something new so that is a proof of sorts but I have not written a proper proof so I need help with that too.

 

Are we ready for a Solution to Collatz Conjecture?

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