Prime numbers of the form $(3^n-1)/2$ 
Problem: Consider the numbers $P_n =(3^n-1)/2$. Find $n$'s for which $P_n$ is prime. Prove that neither $P_{2n}$ nor $P_{5n}$ is prime.
Bonus task: Find more primes $q$ for which $P_{q\cdot n}$ is not prime.

What I have done so far: I have tried $ q=2,3,5,7,11,13,17,19$. So far, when $q=2,5,11,17,19$ and $n=1$ the value isn't a prime. I don't want to calculate any more. So if anyone can help me find the $n$'s for which $P_n$ isn't prime and $q$'s for which $P_{q \cdot n}$.
If you could provide a proof that will help too.
 A: Hint: Let $q\in\mathbb N, q\neq 1$, then
$$ P_q \text{ is composite } \Leftrightarrow \forall n\in\mathbb N: P_{qn} \text{ is composite}$$
In other words, each $q$ you find, such that $P_q$ is not prime (e.g. $q=2,5,11,17,19$) gives you a class $P_{qn}$, which is never prime.
A: There is a unique factor for every value of $n$ in $b^n-1$.  These divide every instance of $b^m-1$ where $n \mid m$.  One can see in base 3, that $\frac 12(3^n-1)$ is written as a string of 1's, eg $11111$.   
When there is a composite number of 1's it can be written with any of the number's divisors, eg $111111 = 11 * 10101 = 111 * 21$   The unique factor that appears with each prime might be calles an algebraic factor of $b^n-1$
This means that the only rep-units that are prime, are those that have a prime number of digits.  Of course, $11$ and $11111$ are both square (2, 11).  
We find, eg 3, 7, 13, in this list, as far as $83$.  This is the limit of UBASIC at the time the table was prepared.
The program i wrote actually tested the algebraic factors in order of size.  The first power is $b^{t(n)}$, where $t(n)$ is the Euler totient of $n$.  The next place is sorted by the number of prime divisors of $n$, and any larger powers.  
There are many more prime algebraic factors, such as this list (increasing size of prime) 6, 3, 10, 12, 14, 9, 7, 15, 24, 21, 26, 36, 13, 40, 60, 33, 46, 72, 70, 63, 108, 132, 86, 130, 154, 371.
A: If you're looking for larger terms,
$(3^n-1)/2$ is prime for $n = 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303$.
See https://oeis.org/A028491.
Note that some of the larger $n$'s may only correspond to probable primes (still almost certain to prime however).
