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The randomness of prime numbers?

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  • bobgillson
    I have often seen in print that prime numbers are random. So from this one could conclude that non prime numbers also have the same property. Yet it is
    Message 1 of 4 , Apr 17, 2013
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      I have often seen in print that prime numbers are random.

      So from this one could conclude that non prime numbers also have the same property.

      Yet it is possible to generate, quite tediously I admit, all the non-prime odd numbers up to any specified limit. Here's a simple program, and forgive my technique as I am not a programmer.

      Let L be the limit of an odd non prime list of integers

      Let n be an odd number greater than 1

      Step 1. Calculate n^2 + 2n + 2n up to the limit L
      Step 2. next n
      Step 3. If n^2 is less than L, repeat Step 1, otherwise print list including 1, excluding duplications.

      Example:

      Let the limit be 100

      So Step 1 produces 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81,87,93,99
      And Step 2 produces 25, 35, 45, 55, 65, 75, 85, 95
      49, 63, 77, 81, 95
      81, 95

      Step 3 produces 1,9,15,21,25,27,33,35,39,45,49,51,55,57,63,65,69,75,77,81,85,87,91,93,95,99.

      All the odd numbers not listed are the primes - so how can they arise randomly from a non random logical process?

      Bob
    • Jack Brennen
      ... Ignore such assertions; even if they are in print, they are wrong. Think about it -- every time you enumerate the prime numbers, you get the same result.
      Message 2 of 4 , Apr 17, 2013
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        On 4/17/2013 10:36 AM, bobgillson@... wrote:
        > I have often seen in print that prime numbers are random.
        >

        Ignore such assertions; even if they are in print, they are
        wrong. Think about it -- every time you enumerate the
        prime numbers, you get the same result. That's the opposite
        of random. :)
      • Phil Carmody
        ... They can t. The people who use the word random in the context of prime numbers either: a) know what they re talking about, and are trying to make a
        Message 3 of 4 , Apr 17, 2013
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          --- On Wed, 4/17/13, bobgillson@... <bobgillson@...> wrote:
          > All the odd numbers not listed are the primes - so how can
          > they arise randomly from a non random logical process?

          They can't. The people who use the word "random" in the context of prime numbers either:
          a) know what they're talking about, and are trying to make a subtle point that you're overlooking; or
          b) don't know what they're talking about.

          The simplest statement that's closest to the truth that ascribes the "random" feature to the primes is probably:
          Given a large enough range of significantly larger numbers (say 10^100 .. 10^100+10^10, for example), the density of primes in that range is as one would expect from a random variable (whose distribution I could state, and which is parametrised by the size of the numbers).

          Primes are to randomness what Pointilism is to brushstrokes.

          Phil
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          [stolen with permission from Daniel B. Cristofani]
        • Chris Caldwell
          ... Indeed, there is nothing random about the primes. You can make a nice mechanical device to find them based on the sieve... What they are trying to say is
          Message 4 of 4 , Apr 17, 2013
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            On 4/17/2013 10:36 AM, bobgillson@... wrote:
            > I have often seen in print that prime numbers are random.

            Jack:
            > Ignore such assertions; even if they are in print, they are wrong. Think about it -- every time you enumerate the prime numbers, you get the same result. That's the opposite of random. :)

            Indeed, there is nothing random about the primes. You can make a nice mechanical device to find them based on the sieve...

            What they are trying to say is bit more complicated, usually one of the following: that parts of their behavior can be modeled using randomness (E.g., Erdos-Kac theorem); and/or, that they often "do" things that surprise us. (E.g., how can something with this much variation come out of a notion as mechanical as the primes?)
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