Visual Prime Pattern identified

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  • #2
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That's awesome. I wish I could have thought of that. I wonder what kinds of applications this could be used in.
 
  • #3
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That's awesome. I wish I could have thought of that. I wonder what kinds of applications this could be used in.

Thank you so much. Its been hard geeting any feedback from this on here. I'm 100% self taught so its hard to get points across when you're not formally trianed. I have the equations behind the visuals but I think there is a way to use collision detection to efficiently determin when concentric circles intersect with evenly spaced parallel verticle lines from which you can decifer every square root, with prime square roots only occuring on the first parabola. Do you see what I'm talking about?
 
  • #5
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wow, amazing article
 
  • #6
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Hi, Jeremy,
I just have one question. Suppose you replace all parabolas by straight lines. That is, no sqrt; the first parabola becomes a line with slope 1 (y=x), and the other parabolas would be replaced by lines with slopes 2,3,4,... (the lines y=2x, y=3x, y=4x, ...). As you draw horizontal lines passing through the marks on your first line (the one with slope 1), would that horizontal still intersect none of the marks on other lines only at prime numbers of the first line?
 
  • #7
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Hi, Jeremy,
I just have one question. Suppose you replace all parabolas by straight lines. That is, no sqrt; the first parabola becomes a line with slope 1 (y=x), and the other parabolas would be replaced by lines with slopes 2,3,4,... (the lines y=2x, y=3x, y=4x, ...). As you draw horizontal lines passing through the marks on your first line (the one with slope 1), would that horizontal still intersect none of the marks on other lines only at prime numbers of the first line?
Dodo,
Yes, all primes P would only intersect on y=1x and y=Px with composites intersecting on thier divisors but you loose your relation to the fourier series and the unit circle which I think are very important.
 
  • #8
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I also find it interesting that the first parabola has a vertex of 1/2.
 
  • #9
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Right; your parabolas do not pass through the origin, instead they have been shifted so that the parabola representing the multiples of n passes through the point in the first parabola that represents the integer n. (This way, the horizontal lines will only intersect true multiples of n, clearing up other instances of n itself.)

A similar thing can be done by shifting the lines I mentioned before; the line with slope n would pass not through the origin, but through the point (n,n) on the first line. Attached is a drawing.

In fact, graphs of any monotonic curve (x^2, x^3, exp x, ln x, ...) would also produce the primes in the same manner (namely, in the manner of http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" [Broken]).

Edit: My bad, x^2 is not, overall, monotonic. I was referring to curves that are increasingly monotonic on the first quadrant; that is, for x>0, whenever y>x you have f(y)>f(x), so that the vertical ordering of the points is preserved.
 

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  • #10
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it seems to me that preservation of order would be intrinsic to any effective sieve. Correct me if I’m wrong but, I don’t think the function I’m using to generate my sieve is necessarily monotonic, although the results can be viewed that way.

where d={1,2,…,x} , z=1-2d/x, n=x/x-2d and y= sqrt((x-d)*d)

tan(acos(z))*n = y (concentric circles intersection with vertical lines)

and factors of y when d=1 where q={1,2,…,y}

y-q^2/2q = 0 mod (1/2) (horizontal intersection of y with vertical lines)
 
  • #11
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This seems like the right place to post this question...

I have been extremely curious about the square roots of prime numbers ever since I had a dream that seemed to indicate there was some sort of characteristics of the resulting irrational numbers. This may not make any sense (as it was a dream, but try to follow what I'm asking), but there was a feeling that the square roots of smaller prime numbers exhibited more "chaotic" behavior in their decimal expansion than larger primes.

If that made no sense at all, I'm simply trying to find some research into the properties of the square roots of prime numbers. I can't seem to find anything on the internet, but if anyone knows of a paper or a link etc I'd appreciate it.
 
  • #12
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Well, there is a sqrt(). Do an experiment: change all your sqrt() to log() in your Flash code, just like that, and then tell me if anything significant has changed. Even better: change all the calls to sqrt() to some function defined by you, thefun(); there you can play with returning sqrt(), log(), or whatever.

I've been skimming through your code, and I'm wondering where are you introducing the tan(acos(z)) part, because I can't find it.

srfriggen: you may want to start a new thread with your question. Personally I don't have an answer, but someone else may.
 
  • #13
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Well, there is a sqrt(). Do an experiment: change all your sqrt() to log() in your Flash code, just like that, and then tell me if anything significant has changed. Even better: change all the calls to sqrt() to some function defined by you, thefun(); there you can play with returning sqrt(), log(), or whatever.

I've been skimming through your code, and I'm wondering where are you introducing the tan(acos(z)) part, because I can't find it.

srfriggen: you may want to start a new thread with your question. Personally I don't have an answer, but someone else may.

Dodo,
I understand the point you are making but the sqrt() is essential in my equation because it perfectly defines the Moiré pattern created by concentric circles and parallell lines. All other functions will miss the intersections of this pattern. My inquiry into this pattern came from an article I read here:
http://www.egge.net/~savory/maths9.htm
harmonics:
http://en.wikipedia.org/wiki/File:Moodswingerscale.svg
the unit circle:
http://upload.wikimedia.org/wikiped...r.svg/1000px-Unit_circle_angles_color.svg.png
and the inverse square law:
http://www.splung.com/cosmology/images/magnitude/inversesquare.jpg
 
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  • #14
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I've been skimming through your code, and I'm wondering where are you introducing the tan(acos(z)) part, because I can't find it.

You wont find the tan(acos(z)) part in my code but I mimic its output.
 
  • #15
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Well, where I was heading to, is that primes are produced because of the sieving process, which in turn comes from the vertical order of the points; and this is not really related to the intersection with the circles.

Leaving the primes apart, you seem interested in the coincidence of the paraboles and the circles, precisely at the lines projected out of the unit circle. I wrote some notes in a PDF that may help with the trigonometry of the situation, and with the reason why the intersections occur precisely at roots of consecutive integers, if that's what you're ultimately asking. The notes also show why that tan(acos(...)) formula is not really right.
 

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  • #16
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Dodo,
Thank you so much for your notes. They made perfect sense to me. I see now that sin(angle) keeps my secant line inside the unit circle with a height of Py = d+1/2 * sin(angle) and my tan(angle) is outside the unit circle with a height of Py = d+1/d+1-2 * tan(angle).


I also see your point about the vertical order of the points. In fact I have a excel spreadsheet with this exact table on it from when I started down this path years ago.

01 02 03 04 05 06 07 08 09 10 11
02[04]06 08 10 12 14 16 18 20
03 06[09]12 15 18 21 24 27
04 08 12[16]20 24 28 32
05 10 15 20[25]30 35
06 12 18 24 30[36]
07 14 21 28 35
08 16 24 32
09 18 27
10 20
11

This ordering is key because it shows the congruence of squares exposing Fermat’s factorization method which is the basis for the quadratic sieve and the general number field sieve. For example look at 36:

36 – 1^2 = 35
36 – 2^2 = 32
36 – 3^2 = 27
36 – 4^2 = 20
36 – 5^2 = 11

I find it more than a coincidence that the simple pattern of parallel lines intersecting with concentric circles produces this ordering exactly showing that primes only have a congruence of square( (P-1)/2)^2 to square ((P+1)/2)^2.

As to your comment that “the sinusoid is a pretty artifact used ONLY to split the diameter on the unit circle”, I have to disagree. Fundamental frequency division produces harmonics. The sinusoid shown is the harmonic produced by dividing the unit circle or fundamental frequency. The intersection of these divisions on the unit circle directly mark the deformation points of the fundamental frequency’s sinusoid when you “mix” the two frequencies (fundamental + harmonic), hence the my comment on the link to the Fourier series and harmonic analysis.
 
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  • #19
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The intersection of these divisions on the unit circle directly mark the deformation points of the fundamental frequency’s sinusoid when you “mix” the two frequencies (fundamental + harmonic), hence the my comment on the link to the Fourier series and harmonic analysis.
To be clear, I'm talking about the orthogonal projection onto the time axis as regards sin with the "directly mark" part here: "intersection of these divisions on the unit circle directly mark the deformation points of the fundamental frequency’s sinusoid"
 
  • #20
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Sorry, Jeremy, but I really don't understand what do you mean. Which is the "time axis" for you, the horizontal axis? What are "deformation points"? If you mean the intersection of the sinusoid with the horizontal diameter of the unit circle, anything I can see is that the diameter is being split in equal parts; I fail to understand where do you see a Fourier series, given that no sinusoids are being added together, or when, for the only sinusoid in sight, the amplitude seems to play no role at all. Is there a calculation involving the sinusoid in one iteration and the sinusoid in the next iteration, and if so, precisely what calculation?
 
  • #21
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Dodo is basically asking to see your equations if you have any. Then everyone can see for themselves their form and what they do.
 
  • #24
695
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Hi, Jeremy,
surely you realize that, in those sites that you cite, sinusoids are being added together. A formula that looks something like this is used,
f(x) = a1 sin(x) + a2 sin(2x) + a3 sin(3x) + ...
where the a1,a2,a3 are the amplitudes (the ones controlled by different slides on those pages).
This is what I fail to see in your drawing, where the sinusoid just stands alone in the middle of the unit circle, and that is why I made the remark about it being used only to split a segment in equal parts.
 
  • #25
204
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Hi, Jeremy,
surely you realize that, in those sites that you cite, sinusoids are being added together. A formula that looks something like this is used,
f(x) = a1 sin(x) + a2 sin(2x) + a3 sin(3x) + ...
where the a1,a2,a3 are the amplitudes (the ones controlled by different slides on those pages).
This is what I fail to see in your drawing, where the sinusoid just stands alone in the middle of the unit circle, and that is why I made the remark about it being used only to split a segment in equal parts.
Oh yes, I definitely understand that and I know my animation does not show the mixing of the sinusoids, it just shows one at a time. What I intend to show is how FFT can be used to identify prime harmonics. A prime number harmonic will only have energy at its frequency and its fundamental (1) across the spectrum, whereas a composite number harmonic will have energy at all its factors across the spectrum. ex: a 1/4 or 4th harmonic of a fundamental frequency will have energy in the 1/2 or 2nd harmonic. Make any sense?
 

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