Strange attractor

[Joshua]

Hey all.

Something great occured to me this morning. I've done a little digging and as far as I know this has not been done yet. I would like to build a strange attractor for prime numbers. I thought one axis would be the distance between prime n and prime n+1. I'm having trouble deciding on the the other two axis and thought you all might have some input.

[Dave]

Hmm. I'm not entirely sure how you'd do this. You might want to think about coming up with some kind of iterative map using primes in there somehow. The problem I foresee is that you're going to have to work with the reals and not just integers in order to create a strange attractor.

[Joshua]

Hm. I like the idea of using an iterative map, but that seems impossible. Or rather, I should say, if I could define n+1 as a function of n (this is what you mean by an iterative map, yes?), if I could do that then I would already be a multi-millionaire! But maybe you had something different in mind.

 

After chewing it over for a while, I think a good question is, "what is the largest distance between two primes?" Not in the strict sense, but in regards to those primes we already know of. Perhaps I should post in a different area to pursue this?

[Dave]

What I mean by an iterative map is something along the lines of:

 

[math]x_{n+1} = f(x_n)[/math].

 

I suppose I should just call it a map

 

I don't really know what you're trying to achieve here; I can't think of any special properties that a strange attractor involving primes might have.

[Joshua]

Very good. That's what I was I trying to say, too. Guess I should just use some images.

 

Your lolly roffle quote kills me!

[ElijahJones]

Your question is a little peculiar because a strange attractor is an attractive fixed set that contains chaotic trajectories of a system of differential equations. These are continuous systems, however you can get discrete systems that are chaotic the quadratic map x->x^2+c is a great example. But to think of primes as being on a strange attractor you would need

 

1) A space containing the prime or whose set of limit points was the set of primes.

 

2) An interative function on that space.

 

Now your last question about the largest distance between two primes is settled by the prime number theorem. Check it out! Eventually if you go far enough out you will find gaps as a large as you like, but the distribution of primes is very complicated and I agree it almost seems random but it really is not. Exactly what underlies the distribution is not understood but Reimann's Hypothesis is thought to strike at the heart of the question. That hypothesis sees primes as sport of intimately married to a certain complex function called the Reimann Zeta Function.

 

I personally enjoy studying primes form the perspective of elementary number theory and Diophantine Equations.