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ObjectivistMathematician

Collatz Conjecture might have been solved

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There is a variation on the Fibonacci sequence present that revealed several different analytic approaches. This is used extensively in the Recursive and Vetting spreadsheets. The read only format provided by google does not make the pulldown list tied to the odd number column to manipulate the exponents and show the successive layering.

You know, from beam analysis and Newton's calculus, that often many calculations are required to determine a precise answer for a specific instance. Trifurcate uses an approach that examines the series of all numbers that 3n+1 requires 3 or more subsequent divisions by two to become odd into three categories while simultaneously working the problem in reverse, one level deeper every time. At each level, 1/3 indicate that it can be reversed no farther. 1/3 show the next step that requires only 1 division to make a 3n+1 become odd again, and 1/3 where 2 divisions are required to do the same.

The same process is applied to the two separate solutions resulting in six groups that are 1 term longer than the preceding, with two stopping, while four groups move to the next stage. Lather, rinse and repeat.

The Fibonacci sequence can be extended indefinitely. It correlates with the odd numbers, which can be extended indefinitely, in such a way as to denote the power of 2 that will reduce the even result of any particular 3n+1 to an odd number again. (n)2x where n is odd allows x to be a variable starting from 0, incrementing by 1 indefinitely. The process working with just the (n)21 and (n)22 in the Trifurcate section can be extended indefinitely. The hardest part to put into words is how the Trifurcate works backwards from a 4n+1 (5, 13, 21, . . . ) to its 3n1 (3, 9, 15, . . . ) counterpart, before being passed off to another 4n+1/3n1 counterpart, without being able to enter at any specific 3nx.

This is the same solution I've seen since September 2013. Now it has been imported into Excel format, and simplified to take advantage of the more limited scope that Google offers online. The variation on the Fibonacci sequence is new. It simplifies the explanation of Recursive Aspect.

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

A refinement of:

 

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Posted (edited)

On 3/1/2017 at 7:23 PM, dream_weaver said:

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.

 

Unfortunately, mere evidence one way or the other counts for nothing in mathematics. There are many statements that were thought to be true until some absurdly large counterexample was discovered.

For example, the conjecture that (n^29) + 14 and (n+1)^29 + 14 are relatively prime for all n, is true for all numbers less than 345253422116355058862366766874868910441560096980654656110408105446268691941239624255384457677726969174087561682040026593303628834116200365400

Edited by SpookyKitty

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Posted (edited)

1 hour ago, SpookyKitty said:

Unfortunately, mere evidence one way or the other counts for nothing in mathematics. There are many statements that were thought to be true until some absurdly large counterexample was discovered.

 

An absurdly large counterexample would be considered as evidence to the contraire, so mere evidence one way or the other either counts for something or it counts for nothing in mathematics.

If you want to check my math, please do. If you have questions pertaining to grasping something you do not understand within it, please ask. I am confident what I have presented is based on sound mathematical principles, although I could use some help understanding why the variation on the Fibonacci sequence works as used.

Edited by dream_weaver

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7 hours ago, dream_weaver said:

I am confident what I have presented is based on sound mathematical principles, although I could use some help understanding why the variation on the Fibonacci sequence works as used.

Is the Fibonacci Sequence the "exponent reducer" in column E of the Recursive spreadsheet?

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Posted (edited)

3 hours ago, New Buddha said:

Is the Fibonacci Sequence the "exponent reducer" in column E of the Recursive spreadsheet?

That is the result of column C + column D, where column C pulls consecutively from column E.

The Fibonacci Series is a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc.

That's why I consider it a variation. If this particular variation has been done before, I don't know what to look for it as.

I also show it in the Vetting spreadsheet, but only to reference which column to look for the result in. It is not used in any of the formulas driving the table between column H and column Z.

Edited by dream_weaver

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In the video, around the 1 minute mark, you introduce n' = (4n+1).  I see that it matches the increments of 8, but I don't follow why this needs to be done.  It seems arbitrary?  Is it something that you noticed could be done, or was there a reason behind it?

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13-5, or 21-13, or 29-21.

It was done to create the set consisting of every 4th term grayed out, and what to do in order to extend it.

 

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I see why the variation on the Fibonacci Series works now. It is another form of expressing the "condensed recap".

 

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On 3/3/2017 at 5:18 AM, dream_weaver said:

 

An absurdly large counterexample would be considered as evidence to the contraire, so mere evidence one way or the other either counts for something or it counts for nothing in mathematics.

If you want to check my math, please do. If you have questions pertaining to grasping something you do not understand within it, please ask. I am confident what I have presented is based on sound mathematical principles, although I could use some help understanding why the variation on the Fibonacci sequence works as used.

 

Any counterexample at all would be considered a proof of the negative, not just evidence.

 

Quote

If you want to check my math, please do. If you have questions pertaining to grasping something you do not understand within it, please ask. I am confident what I have presented is based on sound mathematical principles, although I could use some help understanding why the variation on the Fibonacci sequence works as used.

 

I don't intend to be offensive, but what is there to check, exactly? I'm not at all sure what you think you've achieved here. I don't see anything resembling a proof, and for anything less than that there's no point in checking.

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33 minutes ago, SpookyKitty said:

I don't intend to be offensive, but what is there to check, exactly? I'm not at all sure what you think you've achieved here. I don't see anything resembling a proof, and for anything less than that there's no point in checking.

I have to agree with SK.  I spent about an hour looking at the various spreadsheets and didn't see anything that really resembles a "proof" as it is commonly defined in mathematics.

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Posted (edited)

1 hour ago, SpookyKitty said:

Any counterexample at all would be considered a proof of the negative, not just evidence.

Or, more concisely, it would be evidence of a contradiction, demonstrating conclusively that the Collatz Conjecture is false. To date, no such evidence has been forthcoming.

1 hour ago, SpookyKitty said:

I don't intend to be offensive, but what is there to check, exactly? I'm not at all sure what you think you've achieved here. I don't see anything resembling a proof, and for anything less than that there's no point in checking.

Not to be offensive, then what, specifically, are you commenting on? I've redacted my "QED" for the time being. This is my breakdown of the various indications that show how a 4(n)+1 number can have the Collatz rules applied in reverse, using (2(nx)-1)/3 and (4(ny)-1)/3, until a 3(nz) permits no further exploitation of such a process (Trifurcate Collatz).

Recursive Collatz shows how the odd root of a 3(n)+1 can be predicted with a "Fibonacci" variant.

Vetting Collatz is exploring how the two approaches have some integrative qualities.

Keep in mind the Objectivist )breakdown of possible, probable, certain. In those terms, Collatz is at least possible, and I would warrant even probable. In terms of constructive criticism, what is missing? (Not "Oh my—it's not a proof, in the conventional form of mathematics.) What additional evidence is required to move something from the realm of probable to the realm of certain?

 

I've explored these formulas to the point where they look pretty much self-evident to me. The help needed are questions that lead to avenues that have not yet been explored.

Am I wasting my time on something that is so self-evidentiary true that no further proof is required, or is there some merit in providing something beyond this currant analysis (aside from the obvious fact that 3(1)+1=4=(1)22)) that might prove beneficial to the fence riding skeptics that enjoy the ride over deciding on which side of the fence to step off? (Everything else is 3(n)+1=(n1)2x).

 

 

Edited by dream_weaver

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Posted (edited)

56 minutes ago, dream_weaver said:

(Not "Oh my—it's not a proof, in the conventional form of mathematics.)

Terms such as theorems, proofs, conjectures, etc. are fairly well defined in mathematics.  I happen to believe that much of how modern mathematics is practiced is either Rationalism gone wild or mathematical Platonism.  My use/training in mathematics (Mechanics) is something entirely different from mathematics as practiced by your typical mathematician.  If you ask ten mathematicians about mathematical foundationalism, you'll get 12 answers.

That being said, if you approach something like Collatz Conjecture, then you need to either play by the rules of the game, or reinvent the rules.  The following is a fairly well accepted definition of Proof.

Mathematical Proof

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms,[2][3][4] along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Edited by New Buddha
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Posted (edited)

On 3/6/2017 at 10:47 PM, dream_weaver said:

Not to be offensive, then what, specifically, are you commenting on? I've redacted my "QED" for the time being. This is my breakdown of the various indications that show how a 4(n)+1 number can have the Collatz rules applied in reverse, using (2(nx)-1)/3 and (4(ny)-1)/3, until a 3(nz) permits no further exploitation of such a process (Trifurcate Collatz).

Recursive Collatz shows how the odd root of a 3(n)+1 can be predicted with a "Fibonacci" variant.

Vetting Collatz is exploring how the two approaches have some integrative qualities.

Keep in mind the Objectivist )breakdown of possible, probable, certain. In those terms, Collatz is at least possible, and I would warrant even probable. In terms of constructive criticism, what is missing? (Not "Oh my—it's not a proof, in the conventional form of mathematics.) What additional evidence is required to move something from the realm of probable to the realm of certain?

 

Well, in regard to mathematical theorems, you need a proof in the conventional form of mathematics.

 

Quote

I've explored these formulas to the point where they look pretty much self-evident to me. The help needed are questions that lead to avenues that have not yet been explored.

Am I wasting my time on something that is so self-evidentiary true that no further proof is required, or is there some merit in providing something beyond this currant analysis (aside from the obvious fact that 3(1)+1=4=(1)22)) that might prove beneficial to the fence riding skeptics that enjoy the ride over deciding on which side of the fence to step off? (Everything else is 3(n)+1=(n1)2x).

 

I am going to speak very frankly. You are displaying an astounding amount of arrogance towards the mathematical community. You lack sufficient mathematical background to even understand what the problem is, and yet you speak as if this problem can be solved easily simply by not being a "skeptic" who "enjoys the ride".

To an amateur, such as yourself, it seems as though every problem which can be easily stated can also be easily solved. But this is not at all true. Many great mathematicians have tried very hard to solve this problem, and all of them have failed.

As for your analysis, it is woefully inadequate and misguided. You do not even know how much you don't know when it comes to this problem. It's as if you are trying to build a spaceship out of dirt and sticks.

Now I will say some nice things and constructive suggestions. You seem motivated and smart, and none of what I say above is to dissuade you from trying to actually solve this problem. But in order to even begin and not just waste your own time, you will need the right tools.

First, you need to familiarize yourself with classical logic. Then, you need to read and practice lots of proofs. Third, you need to study lots and lots of discrete mathematics, and since many attempted proofs rely on analytic methods, lots of calculus and lots of complex analysis. Finally, you will also need to study up on computability theory and number theory.

Once you've done all that, study at least the best known attempted proofs. Only then will you even stand a chance.

Edited by SpookyKitty

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On 3/7/2017 at 0:43 AM, New Buddha said:

Mathematical Proof

Visual proof

Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.

Admittedly, what I have painstakingly put together so far sets up a visual way of mapping the Collatz (rearranging the entire number line) into the matrix used throughout that originally led me down this path. Admittedly the approach has been rather unconventional.

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