A wager of £20 on the nature of primes

Factual Questions
1

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:

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: Prime Code - Google Drive

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).

2

I guarantee you that it’s not Mandelbrot’s fractal.

3

I only looked at the patterns, not the math - it looks more like a Moire interference pattern to me.

4

I can’t quite figure out how you go from a table which is basically just
“products of two from 2 to n”,
“all products for three from 2 to n”,
“all products of four from 2 to n”, etc.
…
“all products of m from 2 to n”

to a method of finding primes that are quicker than existing methods. Neither can I figure out what you’re actually trying to represent with your patterns. Your descriptions leave a lot to be desired.

5

Your first hypothesis is reasonably easily proved.

If you take any composite number z, and choose any of its factors that are less than or equal to the square root of the number (x), you know that

z = xk (where k is the product of the remaining factors)
Since k is an integer
z = x(k-j) + xj where j is also a natural number.
So setting j = x
z= x(k-x) + xx
and setting y = k-x, you have:
z = xx + x*y where, x,y,z are natural numbers.
QED.

Clearly if z is a square number y = 0, and if z is prime, x = 1.

The trouble with using alone as a way of determining of a number is prime is that you don’t actually save any time. Finding the value x is simply a matter of finding a factor for z. But if you find a factor, you have already proved the number is not prime.

The converse - generating a list of all the non-primes is no better.

Something you might like to contemplate. The unique prime factorisation theorem:

For any number the prime factors are unique.

For instance - 10 has prime factors of 2,5
100 has prime factors 2,2,5,5
256 has prime factors 2,2,2,2,2,2,2,2
30,030 has prime factors 2,3,5,7,11,13

You can’t find two distinct combinations of prime numbers that yield the same number.
(Obviously ignoring simple reordering.)

And you might like to think about this, and its implication. Just for fun. :smiley:
6 has prime factors 2,3. 7 is prime.
30 has prime factors 2,3,5. 31 is prime.
210 has prime factors 2,3,5,7. 211 is prime.
2310 has prime factors 2,3,5,7,11. 2311 is prime.
30,031 is prime.

The actual distribution of primes is something that has fascinated people for eons. I tend to agree that your interesting patterns are a Moire effect.
You might like to contemplate the Goldbach Conjecture.

And no, what you have is not related to Mandelbot or any other fractal. If you could prove primes as fractal you would be having a chat with the Fields Medal committee.

6

I read (and then reread) this as “A wager of £20 on the nature of primates”. I thought you found some sort of iterative structure in their social hierarchies or something, and was really fascinated and really confused :smack:

7

30,031 = 59x509 …

8

Bugger, it does. I need to check my memory on the proof… It has been a very long time.

ETA - and yes, I got it wrong. It only works for the notional largest prime - you can’t construct them this way.
I was trying to lead into the proof, but didn’t realise it doesn’t work as an example. Such is life.

9

The Ulam Spiral does seem to have fractal-like patterns.
https://www.google.com/search?q=ulam+spiral+rose&client=firefox-a&hs=Xnv&rls=org.mozilla:en-US:official&channel=fflb&tbm=isch&imgil=sO_IbGtsHRIP_M%253A%253Bb5t2eCUQwCr92M%253Bhttp%25253A%25252F%25252F216.171.160.73%25252Fiapp%25252Fulam-spiral&source=iu&fir=sO_IbGtsHRIP_M%253A%252Cb5t2eCUQwCr92M%252C_&usg=__mfK-p-KbnHCVSybXAVZsuiRvLgU%3D&sa=X&ei=AnsCVPeCEsWeggS6t4HQBA&ved=0CDcQ9QEwBA&biw=1920&bih=946#channel=fflb&q=ulam+spiral+&rls=org.mozilla:en-US:official&tbm=isch&facrc=_&imgdii=de7Hi60zh6mZ_M%3A%3BTy7yuxOg2ul13M%3Bde7Hi60zh6mZ_M%3A&imgrc=de7Hi60zh6mZ_M%253A%3B2QonFV5EkoAz2M%3Bhttp%253A%252F%252Fmathworld.wolfram.com%252Fimages%252Feps-gif%252FPrimeSpiralHexagon_1000.gif%3Bhttp%253A%252F%252Fkoozdra.wordpress.com%252F2011%252F09%252F30%252Fhexagon-prime-spiral%252F%3B382%3B325

10

What proof are you talking about?

11

I have not taken a good look at your work so forgive me if I am off-target here.

The most obvious patterns that emerge with prime numbers are the ones shown on various forms of Ulam spirals (previously referenced.) This results from the existence of quadratic sequences that are dense in primes. I am not sure why these exist or if polynomials of other orders exhibit the same kinds of patterns. Nevertheless, a quadratic dense in primes will form a straight diagonal line on a Ulam spiral. The appearance of these intersecting lines at various scales (zoom in and out) is superficially similar to some of the patterns one observes in fractals. I don’t think you could argue that they are actually fractal in nature. Or if they are, the self-similarity of patterns would be observed by zooming out rather than zooming in like one might with classical fractals.

Primes are rich territory for exploring for the mathematically curious. But as far as I know, there have been no patterns discovered that enable us to predict or generate primes by a simple formula. The one exception which I think Francis Vaughan was alluding to is the one that Euclid used to prove the infinitude of primes. That is,

The product of the first n primes ± 1 is coprime to all of those n primes (and is often actually prime).
The fact that this number is coprime but may not be actually prime limits its usefulness severely.
Also since this is a product (similar to a factorial), the numbers generated get exceedingly large very quickly – which again limits its usefulness.

12

On a similar note, there is also Wilson’s theorem for testing primes.

[spoiler]n is a prime number if and only if n divides (n-1)!+1 exactly.

Again not really useful since for any interesting n, (n-1)! is so large as to be uncalculable.
But I am sure that this leads to other theorems that are useful.[/spoiler]

13

So, thus, I seem to have won.Obviously, I want to make it quite clear that I never thought it was madelbrot’s fractal, but I do find the structure curious.

If you would like to go to the same link you can find another program that zooms out- that is really curious.

The idea was (is?) that considering the equation I already have:

n=y**2+xy

If I can find another pattern from n, then in theory I can extract x and y from n- and if x and y reside on the 1x… line then the number must be prime.

I also have an inkling that the theory that a non-prime has to be built of at least 2 combinations of a square and rectangle that is a multiple of the square’s root might help find primes.

For example, if we start with the number 81 (9 squared) and add a number which is not a multiple of 81- say 6.

We can start a table:

81+ 64+ 49+ 36+ 25+ 16+ 9+ 4+

6 23 38 51 62 71 78 83

Considering none of these number a multiple of the top number’s square root- then 81+6 (87) must be prime.

But, anyway, if someone could find another “connection” between n and x and y- then in theory we should have an equation which can test if a number is prime.

Even so, I still find the pattern absolutely beautiful.

14

I am just a layperson when it comes to this type of math, but just wanted to point out to Willmac that in your last post, 87 is not prime as 29 x 3 = 87.

15

Well spotted.

That was a “silly” mistake.Of course, the square rot of 9is 3 and 87-9=78- which is a multiple of 3.

So thus, 87 is not prime.But the system still works.Although I am sure there has got to be a better way .

16

No largest prime.

17

There’s a quicker way to show that 87 is a multiple of 3. 8 + 7 = 15 and 1 + 5 = 6. For any number, if the digit-sum is 3, 6 or 9 (i.e. you do what I just did and get 3, 6 or 9) then the number itself divides by 3 - if the digit-sum is 9 then it also divides by 9.

But keep trying to prove stuff like this - trying and making mistakes is how we learn. Of course, you do run the risk of being fatally bitten by the math bug, and there’s no cure. :cool: #mathteacher

18

This reminds me a lot of something I stumbled across a good few years ago - I was thrashing out ideas related to Fermat’s Last Theorem in my head (in both shapes and numbers - trying to work out whether a series of cubes could be disassembled and reconsistituted into a bigger cube - and I stumbled across what appeared to be a simple method for generating primes.

I wrote some programs to brute-force test it (because I didn’t know any better way) and it seemed to hold up. I showed it to a colleague, who dismissed it as something that everybody already knew, so I sort of lost interest in it.

Later, I tried to describe it on this board and was met with skepticism - if I really had discovered something like that, it would have been a big deal. Unfortunately, I could not remember the precise details of how it all went together and the discussion did not go well.

19

Can you remember any of this?

It would be interesting to look at the overlap.Maybe we might have to concepts that would strengthen both hypothesis.

20

Except for 2 and 3, all primes are one more or one less than some multiple of 6.
http://french.chass.utoronto.ca/as-sa/ASSA-No14/modulus6-spiral90.jpg
How does that figure into your equations?

BTW…on this site down at the bottom, there are two graphs with spiral prime distributions.