--- 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