- --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
>

--thanks, this confirms my own (now computerized) results

> First, you missed an easy one:

>

> 4, 7, 9

>

> Second, the next one seems to be:

>

> (3^541-1)/2, 3^541-2, 3^541

>

> As far as the conjecture about the small examples with double

> powers being the only ones, that would seem to be related to

> the ABC Conjecture.

+example, 1; 2, 3, 5

-example, 2; 4, 7, 9 &

+example, 2; 5, 9, 11 &

-example, 3; 13, 25, 27 &

+example, 4; 41, 81, 83

-example, 5; 121, 241, 243

-example, 541; (large)

where the lines are of the form

+example, n; (1+3^n)/2, 3^n, 2+3^n

or

-example, n; (-1+3^n)/2, -2+3^n, 3^n

all three at the end of the lines being prime-power.

I also have awarded a "&" iff "pack of four."

It claims there are no further examples

for n<=4096.

You are correct that the ABC conjecture looks related to my conjecture that

there are only a finite set of such examples involving TWO or more prime powers.

Weaker conjectures going in same direction are "Pillai's conjecture"

and -- which actually now is a theorem by Mihailescu --

Catalan's conjecture. The success on Catalan suggests to me

that my conjecture might be within reach, although

the proof would, if so, require a lot of effort. - 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. - --- On Mon, 10/1/12, WarrenS <warren.wds@...> wrote:
> Also, at least one heuristic argument

I 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 that

On 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

> failed.

Phil > 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.