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Wrestling with infinity

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  • Robert
    There are an infinite number of primes. Every prime (in fact every integer) can be written in the form k*2^n+1. For k fixed, can it be shown that there exist
    Message 1 of 1 , Sep 30, 2003
      There are an infinite number of primes.

      Every prime (in fact every integer) can be written in the form
      k*2^n+1.

      For k fixed, can it be shown that there exist power series k*2^n+1
      with an infinite number of primes?

      My hunch is that the answer is yes. There are values of k,
      Sierpinski k, where the number of primes is zero, and I think there
      are there are series with covering values after a few primes , i.e.
      with k=1) and can be easily shown to have a finite number of
      primes.

      My studies of k.2^n+1 seem to suggest that the number of primes at
      any given n tend towards a constant, c, where c=p/ln(n). c is
      different for different k, and range from 0 in the case of
      Sierpinski numbers up to about 13 for the very best Payam k. If ln
      (OO) = OO, then p=OO, would seem to be a proposition.


      Note: The problem is that my data is only good up to around n=150000.

      But I am wrestling with infinity and division, which is a dangerous
      thing, I think and I am also dealing with small numbers (<100000
      digits in length) and a formula which might be shaky as I can't
      determine whether c is constant, and that the formula would show an
      ever increasing c or a declining c rather than a limit to which the
      formula is tending towards.

      Any help from you mathematicians here?

      Regards

      Robert Smith
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