Let $n$ be a positive integer.
Conjecture
There are infinitely many prime twins of the form
$$ ( 3^n - 2, 3^n - 4) $$
Examples include
$$(3^2 - 2,3^2 - 4) = ( 7,5 ) $$ $$ ( 3^{37} - 2 , 3^{37} - 4 ) $$ $$ ( 3^{41} - 2 , 3^{41} - 4 ) $$
Notice $2,37,41$ are primes themselves. Coincidence ?
This resembles conjectures about prime twins , fermat primes and mersenne primes. So I do not expect a proof :) I already know the typical statistical arguments why this should be false ( zeta(2) type sums as expected amount ) , so I Will not ask for that either.
My mentor believes this.
How many values are known apart from these ? Are there equivalent conjectures known ?
Is a disproof possible ? I was not able to disprove it. I tried mod arguments.
Any ideas ?