Hello, I am 14.So entirely an amateur.Obviously, I do not want to fall into the, "I am a complete amatuer of your subject but assume my findings will change the world" cliche. Anyway, I recently did some work in correlation to the nature of primes.If you would like to read it, you can access it on the vixra website: http://vixra.org/abs/1408.0087 Anyway, my Dad took one look at the (second half) of my work (particularly the infintely complex and unpredictable pattern it creates) and said "that's got to be mandelbrot's fractal". Admittedly as you zoom out of the pattern it appears that the pattern seems to build itself into bigger patterns- suggesting somewhat a correlation for "fractals" (though probably not mandelbrot's).In fact, when you begin to zoom out of the pattern very weird things begin to occur- bigger patterns working into small patterns- similiar patterns that diverge into chaos. If you would like to make the structure yourself, I have created a python (and pygame) script that generates the structure.Where z equals the disposition of the camera view from point (0,0) and camera is the zoom of the structure(where 0.1 is zoomed in and 10 is zoomed out). You can access the code here: https://drive.google.com/folderview?...kk&usp=sharing I tried to explain that my equations have nothing to do with fractals or complex numbers, but then took the advatage of betting a wager of £20. Anyway, could someone prove him (or me) wrong? Also, could someone (if possible) try to disprove (or encourage) my concepts as the beginning of a method for searching for primes (obviously the square root factors thing has already been proven).
Welcome to PF; Your dad is not the first person to notice a structure suggestive of fractals in prime numbers. I think a closer look usually shows it's just a similarity... any fractal rules stop being predictive after a while. There is a relation between partition numbers and fractals - and thus between the primes the partition numbers are based on and fractals. i.e. http://www.wired.com/2011/01/partition-numbers-fractals/ Did you look at the other approaches to finding primes efficiently that are used when you wrote this paper? Your abstract suggests that the method you used is more efficient than the general sieving approaches and yet there are several. How did you measure this efficiency? note: I'm not sure how work can be in correlation to a subject, though it could be in correlation to another work. I suspect the word you are looking for there is "relation".
Well done, these are great results for someone working on their own. Some comments: I am going to rewrite your first hypothesis as "A positive integer ## z ## can be expressed in the form ## z = y^2 + xy ## where ## x ## is a non-negative integer and ## y ## is a positive integer if and only if ## z ## is prime". This can be shown to be true as follows: let ## y ## be the smallest prime factor of ## z ##, ## w = \frac{z}{y} ## and ## x = w - y ## (note that ## x \ge 0 ##). We then have ## z = yw = y(y + x) = y^2 + xy ##. Of course there may be many other ways ## z ## can be partitioned in this way, but note that for numbers which are squares of primes ## z = p^2 ## the only solution is ## x = 0, y = p ##. I am not sure about your claim about the speed of your primality test: the link to your code didn't work. Note that the simplest primality test only requires testing (odd) numbers up to the square root of the candidate: see this example in python (or here in javascript) - you might like to compare this side-by-side with your algorithm. Be careful with working with large numbers though, note that you can only do exact integer arithmetic on numbers up to approximately 2,1 x 10^{9} on a 32 bit machine, 9.2 x 10^{18} on a 64 bit machine or 9.0 x 10^{15} when using IEEE 754 double-precision floating-point representation which is the way JavaScript represents all numbers. 100,000,000,000,031 is prime but it is not a Mersenne prime. To explore prime numbers further, the Prime Pages are a great place to start.
Hmm... Hmm.. not sure about the link.I think I may have not copied it correctly: https://drive.google.com/folderview?id=0B_Um_csylRd-WHRwb0pvUnNDQkk&usp=sharing Somebody said that there are "several patterns".This is interesting.Considering the concept of simultaneous equations- in theory giving a different equation involving x,z and y one could be given the x and y coordinates of the number- where if the x and y coordinates resided one the 1x1+n line then that number would be prime. Sorry about the code, I am currently working on a video of "zooming out" of the structure which I (personally) find fantastically intricate. Obviously you need pygame installed to run the program... Also... some claims on speed may be incorrect.You have to understand that I wrote large parts of the paper at different times (they're in fact 2 different papers fused togethor) and have not really had any proof reading.Some of the statements I made when I had a lesser understanding of primes.One part I may add is that the resolution effect is considerably more complicated... it seemed initally a bug but as I began massively zoom out I started appearing in structures again. Although this may be because of the nature of my zooming... I am merely multiplying x and y by a given nmber ("camera")- eliminating numbers that are not multiples of "camera".Or, in the case of positive fractions under 1- add intricate detail to the structure.
Z is always a non-prime number. I think, in re-writing my equation, he accidently stated that z is always a prime- when in fact it is the exact opposite.
About your patterns. They have nothing to do with prime numbers, fractals or complex numbers. The reason you see a pattern with repeating characterestics and "hard" edges where the brightness flips from maximum to zero is that you are reducing the values you are calculating modulo 255. Whilst these patterns can be pretty, I don't think there is any underlying significance that makes them worth investigating from a mathematical point of view. By all means continue your investigations, but you should also spend some time looking at the interesting patterns and other features about primes that other people have found.
I think you can collect your £20. I don't think this has anything to do with the Mandelbrot set at all. The lines where you have a sudden change from red to black in your image are, the places where y(y+x) % 255 rolls over from 255 to 0, so this happens when y(y+x) = 256 k, where k is ...., -2, -1, 0, 1, 2, ,,,,,,,, You can only see these curved lines around (0,0) however. They are clearer in this version, where I put the lines further apart If you had an infinite resolution screen, the entire picture would be filled with a line pattern as in the center. The farther away from 0,0 you get, the closer the lines get however, until you can't see the lines at all anymore, but only new patterns created by the jagged lines and the low resolution of the image. This is what is know as a Moire pattern. http://mathworld.wolfram.com/MoirePattern.html You can often see those if you look through 2 fences behind each other, but you can also get them when displaying a series of curves on the rectangular grid of a computer monitor. (note the 3 disk images on the wolfram page, where a series of circles interacts with the rectangular pixel grid) Your prime search program seems to be a version of trial division. You form the set of numbers y(y+x) with x>1 and y>1, but this is the same set of numbers as yx with x>1, y>1 and y>x.
I am aware of this. I am aware of this.I understand that the "zooming out"- is a side effect to the interesting patterns created. Although, if I understand correctly, when you refer to the lines- I believe you talk about something I call "the greater boxes"-built out of the "greater triangles" and the "greater meeting circles". It appears to me that, if you can find a way of simplifying the larger numbers down into a smaller one that resides into the one box at the point of origin. Then, if the number resides within that box, it is non-prime. The problem, of course "is the modulus 255".
Modulus 255 Interestingly I have managed to work around this problem- using a special function I have found a way of representing all of the 255*255*255 colours. This means that I can zoom in incredibly- each zoom providing more fantastic imagery. Obviously I have not got this system completely working- but this is the code I have as of now:
You are still just creating pretty patterns, there is nothing of mathematical significance here and the patterns are still due to the modular reduction ("% num") rather than any feature of the y(y+x) function.
The pretty pictures are pretty Yes, I am (was, partly) aware of this. I just find it curious, I have always found the mundande to others incredibly complex, I have gone was past the age that it is acceptable to stare hours into ants' nests, when flashing past things in obscure country roads- I always feel very depressed that I will never REALLY be able to look out that passing shrub- and, in fact, no one probably ever will. On that behalf, I find many things about this structure fantastic.The greater boxes that adapt in size to the modulus- bringing greater complexity that eventually leads to the smooth curves of the curved lines- which have further lines within them- and probably further lines too. There still, though, is a question relevant to finding primes, or so I think. Considering the equation z=y**2+xy If there is another correlation between x, y and z.Then by using simultaenous equations- you should be able to find out if the number (z) appears on the table, and thus, whether it is prime or not. I have tried many systems, but they all end up simplyfing down to z=y**2+xy. Does anyone have any ideas?
As I pointed out above, this can be further simplifed to z = XY where X = x + y and Y = y. An interest in patterns is a good thing which many mathematicians share. Obsessing about patterns is not a good thing: have you seen the film A Beautiful Mind? There is not in general any other relationship among two factors of a number and the number itself. However this touches on the subjects of perfect numbers and Mersenne primes which may be good areas for you to broaden your interest in number theory.