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Joe Bequette

DOWNEY, CA - USA

Registered: Jul 2005
Posts: 1

Prime Number Absolute Difference Graphic

Hello!

I am not really familiar with the "NKS" or the hard cover/software works of Mr. Wolfram. However I do love to waste time in spreadsheets playing with numbers.

I accidentally fell into the Ulm's Spiral of Prime Numbers, after that experience I decided to try other unique approaches.

The attached graphic file is an example.
One day I created a new spreadsheet in the following steps:
Step 1 - I took column "B" and filled it with the sequence of prime numbers (the first 65,535 to be exact).
Step 2 - In the column next to it, ("C"), at cell location "C3", I generated the absolute difference between each cell from Column "B", =ABS(B3-B2).
Step 3 - I then repeated this absolute difference (relative to the column on it's left), expanding out to the right until I had about 254 columns full of absolute differences.
Step 4 - I then had a ms EXCEL VBA macro program read the value of each cell in the range "C2:IV65536", taking the integer value and color coding the cell to it.
Example:
0 - Black
1 - White
2 - Red
4 - Blue
6 - Pink
8 - Brown
10 - Dark Blue
12 - Purple
14 - Grey

Imagine my surprise when I seemed to generate a "Ying-Yang" composite picture. An image that appears to have chaotic sprawl of hive like clusterings on the left, which then gradually drifts into a skeletal landscape of order on the right. All depending on where the 45 degree median line is crossed in the transition of the progression of the Absolute Difference Columns.

I really do not know if this is the defining evidence of how the universe is constructed based on prime numbers in relation to chaos versus order. However, I do think it is a shadow like image of the informational universe web like structure that surronds, supports, and sustains physical existance.

Regards,

Joe Bequette
finding new ways to waste his free time...

Joe Bequette has attached this image:

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Old Post 07-11-2005 07:36 AM
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Todd Rowland
Wolfram Research
Maryland

Registered: Oct 2003
Posts: 113

phase transitions

That certainly is an interesting looking combination of what is called complex behavior and nested behavior.

Like other observations of prime number behavior, there are some things here which are not as extraordinary as they seem, while there are some interesting things happening.

It turns out that the complex region and the nested region have more to do with the rule, the absolute value of the differences, than anything else. A sequence of random odd numbers produces the same thing as the sequence of prime numbers.

It is an interesting rule, which I haven't seen before. It seems that a random initial condition leads to a rule 60 pattern with two values. Zero and the greatest common denominator. After one step, the sequence of odd numbers becomes all even, except for the first entry. The common denominator in the resulting region is 2, and it quickly becomes 0's and 2's, operating under exactly the Rule 60 elementary cellular automaton.

The greatest common denominator dominates, and that odd first entry, in this case a 1, creates a region which is all 0's and 1's. Note that the absolute difference between 1 and either 0 or 2 is always 1. That is what is on the diagonal, a sequence of 1's.

In this case, the region in the odd phase is nested because it is literally following Rule 60 from a single black cell. See p.58

Since Rule 60 is good at preserving the randomness of its initial conditions, it is probably the case that there are different phases to the absolute value of the differences, corresponding to common denominators. It also obeys a maximal principle, and is like a diffusion, though the number of particles is not conserved. See attached image, which starts with random multiples of 2, followed by random numbers, then by multipls of 3, all between 1 and 120.

One feature of the evolution that is related to the prime numbers is the size of the gaps which seem to correspond to the size of the multi-colored features.

Incidentally, the convention is to display an evolution to a rule by showing the initial condition as the top row. In Mathematica code, your picture looks like Rest[NestList[Abs[Differences[Prepend[#,0]]]&, Array[Prime, 200],200]].

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Old Post 07-20-2005 03:49 AM
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Dan Ellwein
offline
Ridgeland, MS

Registered: May 2004
Posts: 32

this reminds me of the transition that occurs when going from turbulant flow to laminar flow in fluid dynamics...

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Old Post 07-21-2005 04:28 AM
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Anders Larsson


Registered: Aug 2005
Posts: 1

Faster Prediction

If you were able to find a pattern in the graphic representation of the prime numbers you may have found a way to predict primes much faster than todays calculations.

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Old Post 08-18-2005 09:31 PM
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Philip Ronald Dutton
independent
Columbia, SC

Registered: Feb 2004
Posts: 172

primes: unfortunate confusion

The only reason people think about primes all the time is due to the "short cuts" created by the axioms related to multiplication and division. Indeed, I have never seen a definition of the fundamental theorem of arithmetic which did not use the word "product." Isn't a product simply repetitive addition? The main reason primes appear to be so important is because we humans desperately need that "short hand" notation. I would absolutely love to see any of the prominent number theorists rewrite the fundamental theorem of arithmetic in terms of addition and subtraction only! Please! For the sake of man-kind!

Moving along. I would not want to say to an employee, "I will pay you one,two,three,four, five, six, seven, eight,...,nine-hundred and ninety-nine, one-thousand dollars to work for me for one, two, three, four, five, six, seven days." Ultimately, without short-hand, we would count to each other instead of exchange numbers.

Again I plead: Write the axioms down on paper and then erase the ones related to multiplication and division. Now what exactly is a prime number?

The prime number itself is not that special in terms of the number line. A prime number is simply a number which draws attention to the side effect of the language related to the short hand functionality provided by those extra axioms.

Perhaps there is relationship to the form that is found within notions of sequence (ie: the number line).

Maybe one should consider NKS study of primes first by understanding the combination of counting, sequence, and short-hand in terms of NKS.

It seems that the form required for supporting notions of "sequence" are built right into 1D CA. And counting is rather simply with the rule that just moves the "on" cell left or right. As far as notions of "short hand" then the question becomes, "what exactly requires 'short hand?'" This is immediately a meta-mathematical question according to my interpretation. It is based on the application. Our everyday application is exchange of "numerical data" without the requirement to re-count.

It is unfortunate that there are not many good discussions about primes outside of the standard boring and unproductive context of distributions. I could not find any. (really difficult to find good old fashion honest discussion about those pesky numbers in terms of the assumptions and human behaviors behind the writing of the axioms). In my opinion the real meat of the discussion is in the assumptions of the axioms and the interpretation of the language of the axioms and the need for short cuts when counting.

In a generic sense, counting is essentially required for multiplication. Computers could care less about short cuts for addition (figuratively speaking- since there are some cool transistor logic circuits which speedily compute stuff) so why did the humans start to care about the short cuts??? Obviously a rhetorical question (and easy to answer), but I bring it up because it is so easy to forget the pre-tech age human factor.

I think the prime number closet needs to be cleaned out before we should expect NKS to magically predict primes.

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Last edited by Philip Ronald Dutton on 07-20-2007 at 05:47 AM

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