I have a hunch that the language $L = \{ x^n : n \text{ is prime.} \}$ is not context-free. I am trying to show that by contradiction with the Pumping Lemma:
First assume that $L$ is context-free. That means for any string in $L$ of a certain pumping length $p$ or greater, that string can be broken into $s = uvxyz$ where $|vxy| \le p$, $|vy| > 0$, and $uv^ixy^iz$ is in $L$ where $i$ can be any natural number including 0.
I first tried letting $s = x^P$. However, I'm not quite sure how to divide this value up into $uxvyz$ to show that it cannot be pumped. Any advice?
This is not homework. I am practicing on my own. Thanks!