TakenItSeriously

Adding Time to 2D PFHM

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I thought you guys might find this interesting.
 
Adding Time to a 2D Prime Factor Harmonic Matrix to demonstrate the “behavior” of standing vs moving prime factor “waves”
 
Previously I had introduced the Prime Factor Harmonic Matrix which showed that prime factors behaved like waves or specifically 1 dimensional waves that either behave like moving waves or standing harmonic waves within a 2 dimensional matrix of natural numbers.
 
A PFHM is simply any matrix of natural numbers that is dimensioned according to a primorial.
 
When this is done, the pattern of prime factors that make up the primorial behave like standing waves while prime factors that are not part of the primorial would behave like moving waves.
 
In order to better demonstrate this behavior, I added the dimension of time to a 2D 30x7 PFHM which is orthoganal to the plane.
 
Another words I created an animated gif which shows a progression of matrices level by level. i.e. 
210 = 2x3x5x7
  1. 1-210
  2. 211-420
  3. 421-630
  4. ...

ED066DCD-34B7-4AD4-88A9-8C13600065D5.gif.cabd4723019761ef51a945d005a41235.gif

Figure 1: A series of 30x7 matrices of natural numbers such that their prime factors are distributed periodically throughout the matrix. The slot on the left represents the factors of prime number 5, the second shows the factors of 7, the third shows the factors of 11 and the fourth shows the factors of 13. Each frame of the animation shows a progression of levels in the matrix.

Note that since 5 and 7 are both factors of the primorial 210, they never move or they behave like standing waves while all larger prime factor patterns each propagate at a different rate or a different “frequency over time”.

 

 

 
 
Edited by TakenItSeriously

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Just because it’s kind of cool to look at here is an expanded view of the animated 30x7 PFHM stack:

F2B354F6-D7D0-4C09-A653-BEF4127F352B.gif.12d57f892667402964b66c360d37c979.gif

Figure 2: An expanded view of the prime factor patterns.

Above is actually only a partial view of a large array of prime factor patterns in Excel. It starts in the upper left with the harmonic patterns for primes 2, 3, 5, & 7.

Each row actually contains 38 PFPs and the number of rows can extend indefinitely until Excel runs out of memory and crashes. Initially, I tried to include enough prime factors to define the primality up to the millionth matrix (numbers from 1 up to 210,000,000) which requires 1700 prime factor patterns for the primality to be fully defined. Oddly it was at the millionth matrix when Excel started crashing.

The animation represents matrix levels from 10,000 to 10,033 for 33 frames. I chose 33 frames so that the fifth prime (11) would appear to be smooth when displayed in a repeating loop.

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I enjoy the idea of applying physics to Prime numbers. There should be a wave that shows a pattern in Prime numbers. I once posted the idea of having a logarithmic spiral to show a pattern in Prime numbers. I couldn’t get it to work but relating geometry to patterns does things computation can’t.

 

I think the entire problem of finding a pattern in Prime numbers is starting at zero. That is how we count but finding a series is near impossible. Have you ever thought of starting at a starting point other than zero? It may be impossible not to. But I do like your computation and charts. I also like your idea of relating them to physics.

 

I will close with this idea. What if you stop looking at a pattern in Prime numbers and look for patterns in the way they interact with other numbers. For example, I have been trying to solve semi-Primes. If you could prove a number is a semi-Prime, its factors are Prime numbers. So if you take one known Prime number and multiply it by another number if you could prove the resulting number is a semi-Prime, the unknown number is Prime.

 

So what I am saying is that if your charts tested for Primality based on one known Prime and a test value forming a semi-Prime, you would have a pattern. I know this is no easy task. But looking at Prime numbers for awhile now I don’t think a pattern will be formed without somehow placing Prime numbers into a known function, and then find a pattern in that function. Which I think is what you are trying to do with physics, harmonics, and time. I am just suggesting using semi-Primes to see what you can come up with.

 

Also you’ve got to teach me how to create these matrices. And how you are getting those graphics of the patterns. That is just awesome. But I would like you to find patterns where semi-Primes occur in similar charts. You could start with any Prime number. I think it would be less computational.

 

Just an idea. May work, may not.

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I enjoy the idea of applying physics to Prime numbers. There should be a wave that shows a pattern in Prime numbers. I once posted the idea of having a logarithmic spiral to show a pattern in Prime numbers. I couldn’t get it to work but relating geometry to patterns does things computation can’t.

I completely agree that patterns are important in number theory but, in this particular case, rather than looking for patterns indiscriminately and then trying to explain them after the fact, I had first deduced that the harmonic patterns must exist based on the periodicity of prime factors and the resulting patterns had confirmed those deductions.

Quote

I think the entire problem of finding a pattern in Prime numbers is starting at zero. That is how we count but finding a series is near impossible. Have you ever thought of starting at a starting point other than zero? It may be impossible not to. But I do like your computation and charts. I also like your idea of relating them to physics.

Two points:
  1. I discovered an interesting symmetry depending on whether a PFHM starts with 0 or starts with 1.
Try the following excercise:
Take any PFHM starting with 1 and highlight the prime numbers to reveal the harmonic patterns.
 
You will note that a symmetry exists in the x axis with the exception that the first column on the left is a prime column while the second from the last column is a prime column.
 
When we take the same matrix only start it with 0 instead of 1 the assymetry swps such that it is the second column from the left that is prime while the last column on the right is prime.
  1. The second point is that when using a PFHM, one needs not begin at the beginning every time such as with a number seive. That is one of the huge advantages of using a matrix.
You can start with the first matrix, the second matrix or the millionth matrix 
e.g.
for a 30x7 matrix we can calculate the primality for the ranges:
1-210, 211-420, or 209,999,791-210,000,000
without needing to calculate the primality of all preceding matrices first!
for instance here is the primality of the millionth matrix which I derrived directly without first derriving all 999,999 preceding matrices.
8519AD9B-C0AE-4408-9BC2-4ED056CE2EC6.thumb.png.76a7317aedd3d68b9e05df8061bfd4a8.png
 
Note that there may be errors involved since it did require creating a composite of 1,698 prime factor patterns.
 
 
Quote
I will close with this idea. What if you stop looking at a pattern in Prime numbers and look for patterns in the way they interact with other numbers. For example, I have been trying to solve semi-Primes. If you could prove a number is a semi-Prime, its factors are Prime numbers. So if you take one known Prime number and multiply it by another number if you could prove the resulting number is a semi-Prime, the unknown number is Prime. So what I am saying is that if your charts tested for Primality based on one known Prime and a test value forming a semi-Prime, you would have a pattern. I know this is no easy task. But looking at Prime numbers for awhile now I don’t think a pattern will be formed without somehow placing Prime numbers into a known function, and then find a pattern in that function. Which I think is what you are trying to do with physics, harmonics, and time. I am just suggesting using semi-Primes to see what you can come up with.

I’ve been meaning to look into patterns of semi-primes within a PFHM, though I haven’t had the time to look into it yet. I will post any new results regarding semi-primes if I discover any in the future, however, you should note that my first priority at the moment is to find a method for unbounded data compression.

Quote

Also you’ve got to teach me how to create these matrices. And how you are getting those graphics of the patterns. That is just awesome.

Thanks, I appreciate it.

The key to understanding how to create a PFHM are the formulae that are needed to automatically derive the primality patterns within Excell for which there are too many details, in general, to discuss easily within this forum.
 
I may post pics showing the formulae of certain key cells at a later time.
Edited by TakenItSeriously

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On 12/4/2018 at 1:12 PM, TakenItSeriously said:

I completely agree that patterns are important in number theory but, in this particular case, rather than looking for patterns indiscriminately and then trying to explain them after the fact, I had first deduced that the harmonic patterns must exist based on the periodicity of prime factors and the resulting patterns had confirmed those deductions.

 

Don’t let me distract you on your original idea. I know how important it is to pursue your vision.

 

 

I like how your matrices did not have to rely on the previous matrices. I don’t understand the patterns you are using or what your method is. Maybe you could explain in a book format.

 

What interests me now is if you can take my equation and find a pattern between Prime numbers.

 

N^2 = ((((p^2 * N^4 + 2 * N^2 * p^5) + p^8 / N^4) – ((1 – p^2 / (2 * N)))) * ((N^2 / p^2)))

 

I have other less complex equations that will prove p is Prime knowing q and N. The equation is cumbersome, but it will show if 2 numbers are Prime knowing all values.

 

If you are interested in perhaps using any of these equations, I will email you a full set of equations. I don’t mean to distract you from your work, I am just informing you of a pattern you might not otherwise test.

 

My email address is snyder97_bob@hotmail.com

Email me if you are interested in working together. And if you don’t want to team up, I will still send you a copy of my work, because the matrices are your work and I know how it is hard to change one’s ideas when you have a lot of effort put into a project. If you take these equations and prove something it only validates my work. And if you win the Fields Metal in math, I want part of the prize.

 

But I encourage you to write and Amazon Kindle book of how to program these matrices once they are perfected.

 

I want to ask you what is your goal in solving Prime numbers? I will share mine after you share yours. But I am just wondering why we work on such an impossible problem. We probably share a like view. Maybe, Maybe not.

 

Also why do you use Excel instead of other more powerful programs? I admit the Excel computation is impressive. But you did mention it runs out of memory then crashes.

 

Anyway, carry on with your idea and don’t let my equations distract you, but if you ever want to test patterns in semi-Primes as they apply to Primality testing, I am willing to share my work.

 

 

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On 12/6/2018 at 10:12 PM, Trurl said:

I like how your matrices did not have to rely on the previous matrices. I don’t understand the patterns you are using or what your method is. Maybe you could explain in a book format.

Each pattern designates the locations of all multiples for a given prime factor. For example the first pattern designates the positions for all multiples of 2, the next matrix reveals the positions for all multiples of 3 and so on for all given prime factors, for a given level of the matrix.
 
The number of prime factor patterns required depends upon the range of numbers or the level of the matrix that you wish to look at.
On 12/6/2018 at 10:12 PM, Trurl said:

What interests me now is if you can take my equation and find a pattern between Prime numbers.

N^2 = ((((p^2 * N^4 + 2 * N^2 * p^5) + p^8 / N^4) – ((1 – p^2 / (2 * N)))) * ((N^2 / p^2)))

I have other less complex equations that will prove p is Prime knowing q and N. The equation is cumbersome, but it will show if 2 numbers are Prime knowing all values.

Please define your variables and double check your use of brackets as there seems to be some redundancy included.

Edited by TakenItSeriously

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