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19815Re: Find non-integer A such that floor(A^n) is never prime?

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  • David Broadhurst
    Jan 16, 2009
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      --- In primenumbers@yahoogroups.com, "jbrennen" <jfb@...> wrote:

      > is floor(sqrt(18)^n) ever prime? :)

      Heuristically, it ought to be prime for some large odd power n.
      There is no obvious factor in that case and the integral of
      1/(2*n*log(18)) diverges for large n.

      Since there is no PRP for n < 10^4, we might need to go up
      to something like n = 10^4*18^2 to get a half-way decent
      chance of a prime, which would explain the smile sign.

      As for the larger question: I have no good idea for a
      closed form. Did Mills address the question of the
      existence, in principle, of such a number A?

      David Broadhurst
      The Open University is incorporated by Royal Charter (RC 000391),
      an exempt charity in England and Wales and
      a charity registered in Scotland (SC 038302).
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