--- In

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

>

> 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.

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

+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.