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.

On 10/1/2012 9:33 AM, WarrenS wrote:

> If we demand that n, 2n-1, and 2n+1 all simultaneously be prime or prime-power,

> then initial examples found by hand are

>

> n, 2n-1, 2n+1

> 2, 3, 5

> 3, 5, 7

> 5, 9*, 11

> 9*, 17, 19

> 13, 25*, 27*

> 41, 81*, 83

> 121*,241,243*

> (next one over 10^12)

>

> where * for prime powers.

> You can easily see that is is impossible for all three to be prime (except in the

> first two lines) since at least one member of the troika must be divisible by 3. Hence the only way to accomplish it, is to make that one be a power of 3.

> So then we necessarily have an exponentially-sparse set of primes, in that sense resembling the famous "Mersenne primes" 2^p-1 as well as "Proth primes" like 3*2^n+1.

> Another kind of resemblance is the fact that if Q=3^k then

> Q+2 will be easy to test for primality if we know (Q+1)/2 is prime [since then it

> would be a "safeprime"]. In the other direction, (Q-1)/2 might also be easy to test for primality...

>

> Perhaps the 13,25,27 and 121,241,243 lines are the only ones featuring a

> a double (or triple) *.

>

> It seems to me that this is an interesting class of primes.

> Perhaps even more interesting are the "pack of four" cases where n-2 is ALSO prime or prime-power.

>

>

>

>

>

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