Find all $n$ such that $\sqrt{\frac{7^{n}+1}{2}}$ is a prime.
I have tried elementary methods like factorising and using $\pmod{7}$ and so on, but with no luck. I think this question could be approached using Pell's Equation. By trial and error, one can see that $n=1,2$ is a solution. I require a rigorous solution finding all such $n$. Any help would be much appreciated. Thanks a lot :)