<s>[INST] What are the possible prime factors of $3^n+2$ , where $n$ is a positive integer? 
What are the possible prime factors of $3^n+2$, where $n$ is a positive integer ?

It is clear that a prime $p$ for which neither $-2$ nor $-6$ is a quadratic residue modulo $p$, cannot be a prime factor of $3^n+2$, so the primes congruent to $13$ or $23$ modulo $24$ can be ruled out. But is there a simple necessary and sufficient condition whether $p$ can be a prime factor of $3^n+2$ ?
 [/INST] Just an observation not a complete answer.
For a prime $p>3$, applying Fermat's little theorem
$$2^p \equiv 2 \pmod{p}$$
and if there is a minimal $n_0 \in \mathbb{N}$ such that ($p$ - odd!):
$$3^{n_0} \equiv -2 \pmod{p} \Rightarrow 3^{n_0\cdot p} \equiv (-2)^p \equiv -2 \pmod{p}$$
Also $n_0<p-1$, otherwise $n_0=(p-1)\cdot q + r, 0\leq r < p-1$ and
$$3^{n_0}=3^{(p-1)\cdot q + r}=3^{(p-1)\cdot q}\cdot 3^{r}\equiv 3^r \equiv -2 \pmod{p}$$
since $3^{p-1}\equiv 1 \pmod{p}$. So, we can look for such mininal $n_0$ within $\{0,1,2,..,p-1\}$ range.
For example for


*

*$p=5$, $n_0=1$ because $3 \equiv -2 \pmod{5} \Rightarrow 3^{5^k} \equiv -2 \pmod{5}$.

*$p=7$, $n_0=5$ because $3^5 \equiv -2 \pmod{7} \Rightarrow 3^{5\cdot7^k} \equiv -2 \pmod{7}$.

*$p=11$, $n_0=2$ because $3^2 \equiv -2 \pmod{11} \Rightarrow 3^{2\cdot11^k} \equiv -2 \pmod{11}$.

*$p=13$ there is no such $n_0$

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