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Re: n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)

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  • WarrenS
    Also, at least one heuristic argument (involving 1/lnX probability that X is prime) suggests the conjecture that the set of n with n, 2n-1, 2n+1 all
    Message 1 of 6 , Oct 1, 2012
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      Also, at least one heuristic argument
      (involving 1/lnX "probability" that X is prime)
      suggests the conjecture that the set
      of n with n, 2n-1, 2n+1 all simultaneously prime or prime power, is a FINITE set.

      [On the other hand, I can also dream up a different heuristic argument (involving
      sieving the exponent of 3) which suggests it is an INFINITE set! You can place
      your bets on which heuristic to believe...]

      In the former case, it seems reasonably likely that
      Brennan & I have actually already found every example.

      It would be very interesting if anybody could prove this or any similar
      nontrivial finiteness theorem.

      I wondered if such a theorem could be proven under the assumption of the Riemann
      hypothesis & Montgomery pair correlation conjectures, and whatever other standard conjectures about nature of Riemann zeta zeros.
      I made a quick try to produce such a proof, but my attempt failed.
    • Phil Carmody
      ... I can t see an infinite set coming from any reasonable heuristic. You re summing 1/n^2. ... On its own, RH just helps shore up the finite heuristic, as it
      Message 2 of 6 , Oct 1, 2012
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        --- On Mon, 10/1/12, WarrenS <warren.wds@...> wrote:
        > Also, at least one heuristic argument
        > (involving 1/lnX "probability" that X is prime)
        > suggests the conjecture that the set
        > of n with n, 2n-1, 2n+1 all simultaneously prime or prime
        > power, is a FINITE set.
        >
        > [On the other hand, I can also dream up a different
        > heuristic argument (involving
        > sieving the exponent of 3) which suggests it is an INFINITE
        > set!   You can place
        > your bets on which heuristic to believe...]

        I can't see an infinite set coming from any reasonable heuristic. You're summing 1/n^2.

        > In the former case, it seems reasonably likely that
        > Brennan & I have actually already found every example.
        >
        > It would be very interesting if anybody could prove this or
        > any similar nontrivial finiteness theorem.
        >
        > I wondered if such a theorem could be proven under the
        > assumption of the Riemann
        > hypothesis & Montgomery pair correlation conjectures,
        > and whatever other standard conjectures about nature of
        > Riemann zeta zeros.
        > I made a quick try to produce such a proof, but my attempt
        > failed.

        On its own, RH just helps shore up the finite heuristic, as it makes the probabilities better justified.

        Phil
      • WarrenS
        ... --well, I basically agree with you. I can think of a heuristic that says infinite, but I don t like that heuristic :) Meanwhile I happened to notice
        Message 3 of 6 , Oct 1, 2012
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          > I can't see an infinite set coming from any reasonable heuristic. You're summing 1/n^2.

          --well, I basically agree with you. I can think of a heuristic that says infinite, but I don't like that heuristic :) Meanwhile I happened to notice these summaries of immense computations:

          http://oeis.org/A014224
          http://oeis.org/A028491

          http://oeis.org/A051783
          http://oeis.org/A171381 (this last one surprised me!)

          which show that Brennan & my examples are all there are,
          up to 3^195430 at least (wow!) PROVIDED we only allow
          ONE prime-power, the other two need to be genuine primes.
          It would not be hard to use these to genuinely deal with prime powers too,
          but I haven't.

          In the proofs of things like the Catalan conjecture, usually they prove it for very large
          numbers, then deal with the small numbers by computer.
          The fact that these computations have been so immense may make it
          feasible to prove there are no more examples with TWO nonprime prime-powers.
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