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Visual Prime Pattern identified

  1. Mar 18, 2011 #1
  2. jcsd
  3. Mar 18, 2011 #2
    That's awesome. I wish I could have thought of that. I wonder what kinds of applications this could be used in.
  4. Mar 20, 2011 #3
    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. Mar 21, 2011 #4
  6. Mar 22, 2011 #5
    wow, amazing article
  7. Mar 22, 2011 #6
    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?
  8. Mar 22, 2011 #7
    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.
  9. Mar 22, 2011 #8
    I also find it interesting that the first parabola has a vertex of 1/2.
  10. Mar 23, 2011 #9
    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.

    Attached Files:

    Last edited by a moderator: May 5, 2017
  11. Mar 23, 2011 #10
    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)
  12. Mar 23, 2011 #11
    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.
  13. Mar 23, 2011 #12
    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.
  14. Mar 23, 2011 #13
    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:
    the unit circle:
    and the inverse square law:
    Last edited by a moderator: Apr 25, 2017
  15. Mar 23, 2011 #14
    You wont find the tan(acos(z)) part in my code but I mimic its output.
  16. Mar 23, 2011 #15
    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.

    Attached Files:

  17. Mar 24, 2011 #16
    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

    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.
    Last edited: Mar 24, 2011
  18. Mar 24, 2011 #17
  19. Mar 25, 2011 #18
  20. Mar 28, 2011 #19
    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"
  21. Mar 28, 2011 #20
    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?
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