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

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  • jbrennen
    Just a silly little thought I had... It is almost certainly possible to find a positive non-integer A 1 such that for all integer n, floor(A^n) is not prime.
    Message 1 of 6 , Jan 15, 2009
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      Just a silly little thought I had...




      It is almost certainly possible to find a positive non-integer A > 1
      such that for all integer n, floor(A^n) is not prime.


      Question -- does any such number A exist with a simple closed-form
      expression?

      Or, along those lines... is floor(sqrt(18)^n) ever prime? :)
    • David Broadhurst
      ... 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
      Message 2 of 6 , 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
        ----------------------------------------------------------------
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      • David Broadhurst
        ... How about n = 10675 for a BPSW probable prime? It seems that Poisson can be cheated by posting ... and then Poisson s gremlins almost immediately oblige
        Message 3 of 6 , Jan 16, 2009
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          --- In primenumbers@yahoogroups.com, "jbrennen" <jfb@> wrote:

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

          How about n = 10675 for a BPSW probable prime?

          It seems that Poisson can be cheated by posting
          a remark like this:

          > 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

          and then Poisson's gremlins almost immediately oblige
          with a hit, before I can turn off the process:

          ? print(ispseudoprime(sqrtint(18^10675)));
          1

          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).
        • Robert Gerbicz
          2009/1/16 jbrennen ... [Non-text portions of this message have been removed]
          Message 4 of 6 , Jan 16, 2009
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            2009/1/16 jbrennen <jfb@...>

            > Just a silly little thought I had...
            >
            > It is almost certainly possible to find a positive non-integer A > 1
            > such that for all integer n, floor(A^n) is not prime.
            >
            > Question -- does any such number A exist with a simple closed-form
            > expression?
            >
            >

            > A=5+sqrt(19) is good. To prove it use that
            > g(n)=(5+sqrt(19))^n+(5-sqrt(19))^n is an integer sequence for that
            > g(n)=10*g(n-1)-6*g(n-2), using this it's easy by induction that g(n)==1 mod
            > 3 so floor(A^n)=g(n)-1 is divisible by 3 for all n>0 and isn't 3, so not
            > prime. For n<=0 floor(A^n)=0,1 so not prime.
            >
            >
            >


            [Non-text portions of this message have been removed]
          • Chris Caldwell
            Thanks Robert for the nice closed form example. ... No, his note is very short, contains no numerics, and is very focused on his result. However, it would be
            Message 5 of 6 , Jan 16, 2009
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              Thanks Robert for the nice closed form example.

              > 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?

              No, his note is very short, contains no numerics,
              and is very focused on his result. However, it
              would be trivial to modify his methods of proof
              for composites, and of course the exponent 3 can
              be reduced (for composites). CC
            • Jack Brennen
              Ah, great -- that s just the kind of trick I was looking for. Thanks for the example; I m sure it s just one example of a whole family of solutions. Jack
              Message 6 of 6 , Jan 16, 2009
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                Ah, great -- that's just the kind of trick I was looking for.
                Thanks for the example; I'm sure it's just one example of a
                whole family of solutions.

                Jack

                Robert Gerbicz wrote:
                > 2009/1/16 jbrennen <jfb@...>
                >
                >> Just a silly little thought I had...
                >>
                >> It is almost certainly possible to find a positive non-integer A > 1
                >> such that for all integer n, floor(A^n) is not prime.
                >>
                >> Question -- does any such number A exist with a simple closed-form
                >> expression?
                >>
                >>
                >
                >> A=5+sqrt(19) is good. To prove it use that
                >> g(n)=(5+sqrt(19))^n+(5-sqrt(19))^n is an integer sequence for that
                >> g(n)=10*g(n-1)-6*g(n-2), using this it's easy by induction that g(n)==1 mod
                >> 3 so floor(A^n)=g(n)-1 is divisible by 3 for all n>0 and isn't 3, so not
                >> prime. For n<=0 floor(A^n)=0,1 so not prime.
                >>
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