24509Re: [PrimeNumbers] Re: n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)
- Oct 1, 2012--- On Mon, 10/1/12, WarrenS <warren.wds@...> wrote:
> Also, at least one heuristic argumentI can't see an infinite set coming from any reasonable heuristic. You're summing 1/n^2.
> (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 thatOn its own, RH just helps shore up the finite heuristic, as it makes the probabilities better justified.
> 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
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