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The distribution of prime numbers, at first glance, appears to be somewhat random. It is however, deeply structured and deterministic. Does this qualify it as chaotic.

As chaotic is somewhat of a loose term, I will use the following definition: when the initial conditions determine later conditions, but the approximate initial conditions do not approximately determine later conditions. To be more clear, I will break this question down into three parts:

  1. Can particular "conditions" be identified for prime numbers? Again this is hard to answer, but maybe there are similar distributions that prime numbers can be compared to.

This aleads to the next question,

  1. Do the initial conditions determine the later conditions?

And

  1. Do approximate initial conditions not approximate later conditions
Person
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  • There have been interesting results proved by assuming that the primes are distributed randomly and other results assuming exactly the opposite ; that they are very structured and ordered. It all depends on what you observed and want to say. – Patrick Da Silva Feb 29 '16 at 23:02
  • Look up "Ulam spiral" to see that there is definitely some pattern to them. Just not enough for us to exploit when we go look for the big ones. – Arthur Feb 29 '16 at 23:03
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    Chaos is when the present determines the future, but the approximate present does not approximately determine the future. In this sense, one could say that the distribution of primes is chaotic, because there is no known function which approximates the $n$th prime number with decreasing error – Rob Feb 29 '16 at 23:05
  • Same question is discussed already here. – Dietrich Burde Feb 29 '16 at 23:06
  • http://math.stackexchange.com/q/421353/264 – Zev Chonoles Feb 29 '16 at 23:18
  • You need a precise definition of "chaotic" to have an intelligent discussion of this question. – Gerry Myerson Feb 29 '16 at 23:23
  • @RobBland: what? The function that maps $n$ to the $n$-th prime is primitive recursive. No error! – Rob Arthan Feb 29 '16 at 23:45
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    Ok I hope that clarifies my question. I'm starting to realize that maybe it just doesn't make sense to compare a sequence of numbers to chaotic systems – Person Feb 29 '16 at 23:46
  • @Rob Arthan I meant a continuous univariate function which arises from theoretic examination, like how the amount of primes less than $n$ is asymptotic to $n/ \ln n$ – Rob Feb 29 '16 at 23:48
  • @RobBland: but the function that maps $n$ to the number of primes less than $n$ is primitive recursive too. How are you planning to turn "continuous univariate function which arises from theoretic examination" into a definite mathematical notion? – Rob Arthan Feb 29 '16 at 23:53
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    "initial conditions" and "later conditions" are well-defined terms in regard to dynamical systems, but you can't just throw them around as if they make some sense when applied to primes. – Gerry Myerson Feb 29 '16 at 23:58
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    Of possible interest: http://www.ams.org/samplings/math-history/prime-chaos.pdf – Barry Cipra Mar 01 '16 at 00:09

1 Answers1

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As Gerry Myerson already noted in a comment, to talk about the chaoticity of anything, you first have to embed it into the framework of dynamical systems. At the very least, you have to have:

  1. some notion of state of a system (this may be what you understand by condition),
  2. some dependence of future states on present states,
  3. some notion of distance between possible states.

Applying the first point to prime numbers already requires making some very loose associations creativity. There are no different states of prime numbers; they just are.

You could consider an individual prime number as a state of a system and the function $n$ that maps a prime number to the next prime number as your temporal relation between states. But then all you can modify about your sequence is where in the sequence you start – the sequence itself is fixed. The best approximations of a state $p_i$ are its predecessor $p_{i-1}$ and successor $p_{i+1}$. If you insist continuing with this, the best approximations of the present state do allow for the best approximation of the future states. For example, if the present state is $p_i$ the future state after $k$ iterations is $n^k(p_i) = p_{i+k}$ and the respective future states the best approximations of the present state are $p_{i+k-1}$ and $p_{i+k+1}$, which happen to be the best approximations of the future state. But keep in mind that this is already stretching the perspective a lot.

Is the distribution of prime numbers chaotic?

What is that even supposed to mean? I never heard anybody apply the term chaotic to distributions. It’s dynamics that can be chaotic.

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