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July 22nd, 2013, 03:57 PM   #1
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Collatz-Matrix Equations(concept by me)

Well, this is probably thought of already, but might as well give people the concept.

This work is a small fraction of Collatz Theory

So, basically, the idea is that it is an "equation" that lies on multiple dimensions(two dimensional). It is an evolving equation matrix, meaning on an infinite scale it involves infinitely.
Here is the notation of a Collatz Matrix equation:



In this case, the four variables that end with the subscript f are the formulas used in the equation to calculate the matrix. In each case, if the number were to be moved up the matrix, the original number stays at its coordinate while, when moving up the value, the number has to follow the rules of the a_f variable formula. If the number is to be moved left, the value must follow the formula within the b_f slot, if the number were to be moved right it must follow the rule or formula in the u_f slot, and then when moving down the number must follow the rule or formula in the v_f slot. The starting coordinate is the end coordinate in the notation. The x variable refers to the number being used in the whole entire Collatz Matrix equation.

Now, the reason why I developed this is it originally was a game I made, where it follow the same rules. It used some of the rules of the Collatz Conjecture.

Collatz conjecture - Wikipedia, the free encyclopedia

What I found interesting about the idea was the fact that within the Collatz Matrix equation the numbers next to a number in a slot(not in the diagonal directions) were approximately equal. I thought maybe a Collatz Perfect Matrix, which is a matrix following the Collatz Matrix Equation that has values working together equally being equal to the one that is either up, down,left, or right of it.

I still have not answered the question, but the question is:

Does a Collatz Perfect Matrix exist?

This question probably isn't important, so I just want thoughts on this concept.

EDIT: If your a bit confused, here is a link to the game: Collatz's Matrix game

EDIT2: Also, I forgot to mention, the question or problem only applies for Collatz's rules, no other rules.



X in the formula rules refers to the evolving x value, while the x next to the C refers to the initial x value.

And the question also refers to a matrix that and
EDIT3: I think I messed up writing it down. This should be the correct one.
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July 22nd, 2013, 04:02 PM   #2
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Re: Collatz-Matrix Equations(concept by me)

It seems I can't edit, so here is the fix to the latex parts in order:





and
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July 22nd, 2013, 04:10 PM   #3
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Re: Collatz-Matrix Equations(concept by me)

Here is an Open Office Math file to explain the parts of the concept: https://www.dropbox.com/s/n8o6dauuxq...z%20Theory.odf

I made a Collatz-Matrix algorithm that can take a prime product and break it up into its two different factors. Right now, it is inefficient because there is currently no equation formed yet to retrieve the values of specific coordinates related to the value inputed. Also, the increase in the size of the prime product affects how large the matrix solutions must be in order to retrieve the prime factors involved in the prime products. Another inefficiency is the size of the matrix solutions at the moment.

However, if the equations can be found to describe the values within the matrix solutions this maybe a new find of breaking down prime products.

EDIT: For example, for the sake of simplicity, let us take the two prime numbers 13 and 3 and multiply them together(the product is 39). Now, the algorithmic equation would look like the following:



Here are the solutions:


In many of the solutions, one of the prime factors comes up once or twice. This means this will take less time to find the prime factor within the "jumble" of numbers that have resulted from the solutions. With my Collatz determinant equations, I may be able to form an equation that will predict the locations of the factors within the matrix solutions. If so, this may be revolutionary.

Another thing to do to clean up the mess would be to set up a set that consists only one of each value from all the solutions(and then only include prime numbers into the set). And then you could eliminate all values that are not prime.

Here would be the result:


Primes larger than the initial value also can be eliminated:


And one could run through these possibilities to test for the factors.

There is also something I need to clarify.

The zeros in the solutions do not necessarily mean they are 0(I know it is kind of paradoxical, but for the sake of argument). Though 0 can refer to the fact that element was a result of the parameters within the equations, the 0 can also mean a null-zero element(meaning it is non-existent as a part of the matrix solution). More definition will be provided later.

Another thing I forgot to add, which is the Collatz-Matrix Prime Factorization algorithm.
Here is problem to be addressed by the Collatz-Matrix Prime Factorization algorithm.

Since, by prediction, the size of the matrix solutions will increase as the size of the composite number gets larger the amount of matrix solutions that are produced by the Collatz-Matrix equation will increase because an increase in the size of the matrix solutions will bring a larger amount of solutions available(though there are exceptions). However, the problem is still pending because I haven't measured to a point where if the ratio of the size of the number to the size of the matrix solutions to the amount of matrix solutions will be linear, exponential, or a wave function.








" />
" />
_{1}p_{2}" />
_{1}p_{2}" />






Where the function for these proportions is defined by



I am still working on the function, but this function is defined as such. One to keep in mind is this function does not refer to the proportion between the size of the matrix solutions and how many that will occur, but the composite and the minimal size of matrix solution needed, where k = d, to find the factors of the composite.

Though, here are a few questions that must be asked:

1. Is the relation between the size of the matrix solution and the number of matrix solutions exponential, linear, wave-like, all, or none?

2. For the Collatz-Matrix Prime Factorization algorithm, is the relation between the size(or magnitude) of the composite number and the size of the matrix solution linear, exponential, wave-like, or neither?(If neither, then the function of does not exist). Simply put, the question asks whether there is a function to describe the relationship between and .

3. In relation with the first question, is growth dependent on the initial position of x?

I forgot to mention that x refers to the size of the composite made up of two primes and two primes only, where a composite is referred to as , where n is the index of the composite number made up of two primes.

For example, 4 maybe the first composite made up of two primes. So, (This is not confirmed, it most likely is but I still have to check).







Now, using the proportions given above, here is the result:






Here is another, which is 6:





Now, using the proportions given above, here is the result:






Here is a problem, however, that arises from proving the second question, whether it is yes or no.

Since with function there requires an introduction of which represents a composite of two primes, where n represents the index of such a composite, and since there are infinite primes that exist, which is proven by Euler: Euler?s analytic proof of the infinitude of the primes | Beyond the Numbers(This is just a source out of many that give the article of his proofs), there would be no known way to make an analysis of the growth of the matrix solution sizes with the composite numbers of two primes.

Since there are an infinitesimal set of prime numbers, would be hard to be proven to exist because there is no known way(as I know of) to prove that an exponential, linear, or wave-like growth in the relationship between and exists. The only way to prove such a relationship would be to use Calculus(I think) and the concept of infinity in some fashion.

EDIT: I made a mistake. The function would only be until there is a way describe it, being exponential, linear, of wave-like. The n would represent the n in , where is the composite in the index of n. would actually be referred to as a semi prime, as I was recently informed.

Instead of looking at each semi-prime in order, it seems more consistent to observe each prime factor with another prime factor that is larger than it. For example, the first prime to analyze would be 2. Now, the tricky thing with this function is since there are infinite primes in analyze with the prime 2 this causes a problem with trying to analyze the primes precisely.

where is the normal sequence of prime numbers.

Then .





If the function can be found, an equation can be formed to predict the amount of solutions for the Collatz-Matrix Prime Factorization equation because the amount of matrix solutions is dependent on the size of the matrix solutions. This can also be used to predict the efficiency of the equation.

The following, if this is true, would be the equation for finding the area of a matrix solution from a Collatz-Matrix equation(this only applies to Collatz-Matrix Prime Factorization equations).



Which is also...





Here are a few more equations that can be formed, though they are irrelevant(at the moment).





One thing I noticed while graphing this function is how as the semi-primes get larger for the Collatz-Matrix Prime Factorization equation the more primes that occur through out the matrix solutions that exist before the prime numbers that make up the semi-prime within the equation.
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