(Note the overline denoting the complement.)
I know that, for every $k \geq 2$, the language $L_k = \overline{\{w^k \mid w \in \Sigma^*\}}$ is context-free. The proof is a simple generalization of the proof of the $k = 2$ case given here.
So I'm wondering if $L = \overline{\{w^k \mid w \in \Sigma^*, k \geq 2\}} = \bigcap_{k \geq 2} L_k$ is context-free?
Words which are not a proper power of a smaller word are called primitive. As far as I know, context-freeness of primitive words is an open problem, which has resisted years of research.
There is a book on the subject, Context-Free Languages And Primitive Words by Dömösi and Ito (2014).
A word is said to be primitive if it cannot be represented as any power of another word. It is a well-known conjecture that the set of all primitive words $Q$ over a non-trivial alphabet is not context-free: this conjecture is still open. In this book, the authors deal with properties of primitive words over a non-primitive alphabet, the language consisting of all primitive words and related languages.
