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Collatz Conjecture might have been solved

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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. 

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I discovered that if the 12n+10 could be combined with the 12n+4 to become 6n+4, a point you indicated you had derived the two from. Did you happen to notice that the 6n+4 column is what you would get from 3n+1 applied to the odd numbers, sequentially?

 

It is to the 6n+4 set that the modified Fibonacci sequence applies.

 

FWIW, I had to take Conway's talk on "Unsettleable Sequences" with a grain of salt. Is this the same Conway associated with the computer screensaver program called "The Game of LIfe"? Hmmm.

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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.

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Well, y'alls maths looked to be similar to work I have seen before yet I wonder if you two accomplished any advances?

 

Anyone find the suggestion I made worth a second consideration?

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Indirectly, the video presentation came from seeing your various length chains in post #48. I had noticed the pattern before, and made some headway by a "brute force" approach. The video demonstrated to me how it could be systematized and extended as required. Just_a_hobbyist is better with the charts, and had a little feedback from some of the folk over at mymathforum.

Edited by dream_weaver

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That's how it has gone for the past 20 years. 

 

I really think trying to prove the number to next to one is not the way.

 

Y'all probably have better maths skills.  Can we show that given any number not in the cycle that the value descends towards zero over the distance?

 

Once we can show the X/2 wins over the 3X+1 and we show that there is a cycle on the value One we can show that once the value of our x becomes one of the cycle values that it gets stuck in cycle and not that it "Goes to One."

 Proof of this is that for any 3X+Y where Y is odd there is a cycle of the Y

 

We must prove for all Odd Y not just Y == 1

 

I understand conformity is a powerful influence.. How can everyone be wrong for so long right? Well no one has a solution who has conformed now do they?

 

Take a walk on the wild side here.

 

What maths shows the over all value through iterations in [3X+Y,X/2] <-- my notation for cycle here

 

I mean 3X+Y and the value goes up then X/2 and the value goes down. It seems obvious to me X/2 wins out over the distance. I just need some help with the math on that.  The cycle is obvious.

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You bring conformity in. At the beginning of my investigation, I hadn't looked at any other work on the matter. I opened an excel sheet and tried different things.
It was only when I went to organize it to present, that I pulled Cody's proof up. I did not even examine the proof. I went with his treatment of the odd vs even numbers.

It went well with a matrix I had set up.

 

After going to the math forum, I got several sets of eyes on it that provided me with feedback. I had not accomplished what I thought I had set out to do. I asked questions to try to understand where it fell short.

 

Now I've added two more sections that, again, I have not seen elsewhere. In my mind, it's unquestionable. I see it too clearly. But that is not proof. Proof is organizing this information in a step by step analysis and explanation of what is going on with it, where another person, with sufficient knowledge of the subject can concur.

 

As one commenter mentioned, is it any surprise that so many approaches bear resemblances to one another when seeking the same resolution.

 

One other aspect that weighs in to me, is the how far mathematics has divorced itself in many areas from connecting their concepts back to the perceptual level. After examining Fermat's theorem from a more ancient greek approach, it became clear to me why Fermat was correct. I doubt I could ever get that from examining Wiles 108 page paper. I have to weigh the state of the community against the time and money I would have to spend to hire a tutor.

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I think it's simpler then we have assumed.  Collatz may have had it wrong.

 

Let's go with the http://en.wikipedia.org/wiki/Occam%27s_razor approach.

 

Here is where I can stand some help and clarity.

 

How can we show the equation-proof of relative value descending towards zero?

 

Given some integer we apply the [3x+1,x/2] correct so it "grows" in value if odd and it "shrinks" in value if even.

Now what is the mathematics that explains the decent towards zero? I mean it's obvious that the value of X does reduce over a distance right?

 

So what maths do we write to show this relationship between the two functions of { 3X+1, X/2 } ???

 

You see if we can show it always reduces over distance then to explain why it cycles on the 3X+odd is obvious too and so the function simply '"steps through" descending towards zero in value over distance until it "hits" one of the cycle values.

 

 Remember Collatz was studying the Cycle of  1/3 when he made this famous conjecture.

 

So what maths works here?  How can we define the overall reduction in value over a distance for [3X+1,X/2] ???

 

 I do think this is the correct direction of "proof" and that the "proving the path to One" is a "Red Herring."

 

Anyone agree or not?  Reasons??  Please share. 

This could be much simpler then we have thought.

Thus it is a reduction in value over distance separate from the cycle generated by the odd addend.

 

Ernst

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I was not aware Collatz was studying the Cycle of 1/3. I was drawn in by the allure to try to understand why this particular conjecture works.

 

Occam's razor relies on simplicity too much for me. I'm looking for essential characteristics that I can find and identify as I examined the conjecture.

 

Considering the conjecture asks, in essence, why do all numbers reduce to one when subjugated to this two pronged if/then - it is indeed what is asked to be proven. 

 

Keep in mind, Keplar worked under the assumption that Brahe's data should be able to be laid out with circles somehow. It was not until he relinquished that notion that he was able to work with that data differently, and only then discovered the elliptical nature of the planetary orbits.

 

Starting with the set of odd numbers, you can multiply each element of the set by 2 and not get all the even numbers. Only by taking each subsequent set in succession and repeating the process, will you get them all. This is one method by which we can know that all even numbers, divided by 2, ultimately reveal the odd number which lies at their root.

Edited by dream_weaver

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I'm giving thought to how to formulate the transit of value and I welcome any help with that maths.

 

The obviousness of the cycle of the Y of 3X+Y hasn't been included in what I have read from others attempting proof.

 

So, I think this is the right direction for a proper proof.  The "Goes to One" may be a misnomer. Should say has a cycle on One or a cycle on the Y of 3X+Y

 That divides the elements into two sets. Those that are part of that cycle and those that are not.

 Of those two sets there is only one common element to both 3x+y and x/2 and that is the first even after the 3x+y

 

So, cycle of Y, all others are transit values that decline in value over distance. One element in common between the two functions in the algorithm of if odd or if even.

 

 

That's what I got out of the past 18 years of working with this.

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The best help I got was over at MyMathForum. Albeit, if you don't use LaTeX and concise nomenclature, they may not take you very seriously. They helped point out some shortcomings in my early work, prompting me to re-examine it.

 

A tutor from the tutoring agency contacted me, and I'll be busy for a while explaining and and getting translated what I have into conventional math symbols.

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I see...

 

After all the time I have worked with this I have little doubt.  It's the simplest and it's all true.

 

Okay, if you will refrain from deleting this post; Anyone interested can contact me at [email protected]

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For personal reasons, I've just decided not to pursue The Collatz any farther.

 

I think Paul Erdős (the most prolific mathematician of the past century), captured it pretty well having said that the world of mathematics is not yet ready for problems like this.

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Not Ready? Well, not really because we humans are basically the same humans genetically as those that lived 2000 years ago so we humans can understand the answer. What happens is that we resist change.  

 

Science is rife with examples of good minds balking at change like Tesla slamming Albert Einstein and Edison slamming Tesla.  Just look at what crap Cantor was subjected to and the most famous in my mind was http://en.wikipedia.org/wiki/Giordano_Bruno . I have experienced "The Slam" first hand when I introduced Dynamic Unary to the world, before the paper was on arXive and even before the subject was relieved. A well respected Data Compression person wrote a reply to my announcement that I would be sharing something with everyone soon that flat out stated that my paper would never be read by anyone and that I should accept that as the way it is.  Looking back now I wish I had replied that his "Freudian slip" was showing and I could see his cute little agenda but I was polite and wrote that I would be thankful that it was on arXiv if nothing else.  Dynamic Unary generates dynamic integers by the way so they are bonafide mathematical objects  just like ordinary numbers are,.

 

So I disagree with Paul Erdős on the grounds that we must first believe in the change before we can change.

 

I restate that the Collatz is just one of an infinite set of systems (dynamic equations) which are differentiated by the addend in the function 3X+Y where X is the current value of state and Y is the odd addend.

The proof of this is in the parity language structure where if the 3X+Y is represented with one parity and the X/2 is represented by the opposite parity. Through iteration  then a relationship to the Fibonacci series is observed because no consecutive 3X+Y functions occur. Thus although the sequences of values are different for the same X but different Ys' the common parity language structure is proof that all odd Y belong to the same set or best said 3X+1 belongs to that set of all odd Y

 

So, we must change our perspective here and that therein is what I feel Paul is really saying and not that we are too dumb to figure it out and somehow the humans of the future are somehow smart enough. 

 

Changing talking points here, I have written many C language programs using these dynamic equations so they are very useful and of mathematical significance. Therefore "We are indeed Ready for it."

 

For anyone interested in Cryptography I point out that you can encode and decode data using this Parity language.  So there is a Cryptography application right there.  Been there done that....

 

 Again "We are Ready for it." We just need to think in terms of iterations(dynamic) and not in just the linear (A+B=C) mathematics. The Linear I think is the approach most have taken in trying to produce a solution to the Conjecture.

.

 The answer already exists so it's on us to change in order to see it So.. Keep on Banging those Rocks Together my friends.  http://en.wikiquote.org/wiki/The_Hitchhiker%27s_Guide_to_the_Galaxy

 

 

Ernst

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It occurred to me that I should write a paper on this subject.  I ask of you, those of you that have read what I have posted in this thread, to let me know if any one else has written these things down.  I do not wish to repeat prior Art.

 

Also Greg, Are you done discussing the Collatz as well?  For the rest of your life or is it that what you wanted to do do didn't pan out when you talked to that tutor?

 

 I'm thinking of making the question of "Did Lothor Collatz" get it wrong.  What do you think?

Edited by Ernst

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The interview with the prospective tutor went quite well. I've parsed Collatz to the limits of my understanding at this time, and have bettered my understanding of the conjecture in the process. I see more that can be done with what I've got so far. Time has been in short supply for pursuing it right now. If you've specific questions regarding my findings to date, I can try to elaborate on them.

 

I don't think Lothor Collatz got it wrong, but I do find set theory cumbersome to work (converting what I have into conventional symbols) with as I've encountered it thus far, and it is so intertwined with the 3x+1 conjecture, I see it turning up in just about every step I've developed.

 

I've lost track of how many hours I've put into it. I know when I've had some breakthroughs on it, I've given it 20+ hours a week until I've been satisfied with my grasp of the math involved.

 

The Paul Erdős citation fits in nicely with something you said.

"We just need to think in terms of iterations(dynamic) and not in just the linear (A+B=C) mathematics. The Linear I think is the approach most have taken in trying to produce a solution to the Conjecture."

 

I don't know about the iterations or dynamic aspect, but I agree that the solution is not linear. Seeing "Common Core" approaches to math problems posted by parents trying to help their kids with math, and JohnR over at MyMathForum, who seems knowledgeable about math, but unable to easily read what I presented, is in part why the Paul Erdős quip resonated with me.

 

 

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Science is rife with examples of good minds balking at change like Tesla slamming Albert Einstein and Edison slamming Tesla. 

 

Edison opposed using alternating current in homes because he knew such current is far more dangerous than direct current.  Though Edison invented the idea of distributing power using high voltage, low current rather than the reverse, he didn't think the higher voltages enabled by using alternating current justified the decrease in safety.  (He was wrong in that people were willing to put up with greater risk in order to get cheaper power.)

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Thanks Mark for that addition.  The conflict between Edison and Tesla is classic.

 

Guys I will be digging out my attractor finder program to generate a new list of all the cycles of the [A(X)+Y,X/2] since I cannot find the old list.

I remember that there is a single cycle for all odd Y addends that are Powers of three such as the famous Collatz where Y=3^0

 I wrote that program 14 years ago. My my how time goes by when one works on this challenge.

 

 I can generate all the cycles into the millions of odd Ys'  with the Workstation I now have so if anyone wants are really long list of all the cycles let me know otherwise I will cut it off at about the first 10,000 ys'.

 

I will post it on my site and if I can figure out how to work the download add-on I can make it a downloadable file otherwise there will be as many as the editor will hold.

 

@Greg No specific question at this time I simply didn't want you to feel left out of the group if you are not actively working on it.  "Collatz friends" are important to me.

 

Oh and what I was thinking of doing is only applicable to Ys that are powers of three so that is not a global solution although we can divide the input into more than one set.

If you are interested in a quick proof of more than one cycle simply work with 3x+5 with input set {1,5} you will see (8 4 2 1) and ( 20 10 5 ) Also Cycle notation is different from set notation as we do not use commas in cycle notation.

Edited by Ernst

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You know I see that the last version of the Attractor Program here is dated 2005 so I am wrong about how long ago it was.  Also I see that I didn't have as grand a format as I remember???  I guess I spent time editing the output.  So, I'm thinking I should rework to take advantage of multicore programming.  However, I am going to cut and paste the output here so my Collatz friends can see that there are an infinity of cycles and that only addends that are a power of three have a single cycle and the rest have more cycles.  Therefore there are many "sets" the inputs divide into.  Obviously I present a big clue about this Collatz Conjecture in that I am sharing the bigger picture.

 

Attractor Finder program by Ernst Berg June 2005
Report for the Y of 1 is a power of 3 and a single attractor system
Report for the Y of 3 is a power of 3 and a single attractor system
The attractor >> 1 has been added. Run had 5 steps and the X is 1
The attractor >> 19 has been added. Run had 18 steps and the X is 3
The attractor >> 5 has been added. Run had 4 steps and the X is 5
The attractor >> 23 has been added. Run had 9 steps and the X is 23
The attractor >> 187 has been added. Run had 47 steps and the X is 123
The attractor >> 347 has been added. Run had 44 steps and the X is 171
 Report for the Y of 5
--------
The value of Y is now 7
The attractor >> 5 has been added. Run had 9 steps and the X is 1
The attractor >> 7 has been added. Run had 4 steps and the X is 7
 Report for the Y of 7
--------
The value of Y is now 9
Report for the Y of 9 is a power of 3 and a single attractor system
The attractor >> 1 has been added. Run had 9 steps and the X is 1
The attractor >> 13 has been added. Run had 28 steps and the X is 3
The attractor >> 11 has been added. Run had 4 steps and the X is 11
 Report for the Y of 11
--------
The value of Y is now 13
The attractor >> 1 has been added. Run had 6 steps and the X is 1
The attractor >> 13 has been added. Run had 4 steps and the X is 13
The attractor >> 131 has been added. Run had 43 steps and the X is 19
The attractor >> 211 has been added. Run had 26 steps and the X is 99
The attractor >> 259 has been added. Run had 13 steps and the X is 123
The attractor >> 227 has been added. Run had 16 steps and the X is 147
The attractor >> 287 has been added. Run had 21 steps and the X is 159
The attractor >> 251 has been added. Run had 16 steps and the X is 163
The attractor >> 283 has been added. Run had 14 steps and the X is 283
The attractor >> 319 has been added. Run had 14 steps and the X is 319
 Report for the Y of 13
--------
The value of Y is now 15
The attractor >> 57 has been added. Run had 20 steps and the X is 1
The attractor >> 3 has been added. Run had 5 steps and the X is 3
The attractor >> 15 has been added. Run had 6 steps and the X is 5
The attractor >> 69 has been added. Run had 11 steps and the X is 41
The attractor >> 561 has been added. Run had 49 steps and the X is 241
The attractor >> 1041 has been added. Run had 46 steps and the X is 337
 Report for the Y of 15
--------
 

 

Simply change the Addend to the current Y value and run that with the distinguished X value to verify the data.

Edited by Ernst

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So, how many of you knew about the multi-cycles already?

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Ernst,

 

There are many patterns which can be discovered examining the Collatz. For instance, one of the parameters Collatz set up is if the result is odd, multiply it by 3 and add one.

Since the ancient Greeks it has been know that odd time odd is odd, while even times odd or odd times even and even times even all yield even results. Adding 1 to an odd number makes it even. Even numbers are divisible by 2.

 

If you multiply the set of odd numbers by 3 and add 1, you end up with about 1/2 of the even numbers.

Divide them by 2 and you will discover that the following pattern emerges.

 

2. 1, (2+a), 1, 2, 1, (2+B), 1, 2, 1, (2+c), 1, 2, 1, (2+d), 1, 2, 1, (2+e), 1, 2, 1, (2+f) . . . being the number of times the even result can be divided by 2 before becoming odd again.

a, b,    c,     d, e, f, . . .

 

Doing this, you can discover that the results of the 1, 2, 1 re-subjected to (3n+1) eventually result in a (2+?).

 

The question that remains open is: Are these facts significant or relevant toward establishing a proof?

Edited by dream_weaver

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I am considering reworking the "Attractor Finder" program I wrote back in 2004 and publishing something of a list of cycles of the dynamic system [3X+Y,X/2] is there any interest here in advancing our understanding of these dynamic systems? We simply must expand our conceptual foci.

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Posted (edited)

Nearly two years later, I find that I created a document on November 24, 2014 and finally made it public on February 18, 2017. Since then I made two more public on February 28, 2017.

These three google spreadsheets have condensed the range and scope of an excel spreadsheet immensely.

It is too soon to expect any serious comment on these works, and I've decided to bypass the "gatekeepers" in the ongoing effort to continue to examine the facts, as I understand them, for myself.

For anyone interested in the conjecture that are not following the limited (but presumably interested) audience I've submitted this to so far, there they are:

1. Recursive Collatz Aspect
2. Trifurcate Collatz
3. Vetting Collatz

1 and 2 isolate two very distinctive natures I've observed in my examinations, so far, as identified. 3 puts the two approaches together as I currently understand it.

I'm not so much looking to examine other approaches to the Collatz, but rather I am interested in where there may be shortcomings in the approach provided.

Or to put it in Objectivist terminology: with each formula, is it True or False?—Right or Wrong? with the same questions being applied to each consecutive page.

The underlying premise contained is operating from the standpoint that all of the evidence points to the Conjecture being true, with absolutely no evidence available to the contraire. The onus of proof lies on he who asserts the positive. A claim or assertion that there may be an unidentified, unsubstantiated exception only holds traction with skeptics seeking a foothold in the realm of unmitigated doubt. The positive, in this case, is laid out in "spades", to the best of my knowledge.

Edited by dream_weaver

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Posted (edited)

Per this video, he suggests that the solution to the Collatz Conjecture might be tied to the Halting Problem.  It seems that an effective solution would be one that involves an algorithm that could tell the number of steps it will take for any number to reach 1.  How were you approaching the problem?  Can it be easily summarized?

 

 

 

Edited by New Buddha

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