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24509Re: [PrimeNumbers] Re: n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)

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  • Phil Carmody
    Oct 1, 2012
      --- 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.

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