I’m currently learning for my exams this semester and tried to solve some old exams from the last years.

The question is to show, that L ist undecidable. $ L=\{w|T(M_w)\neq\emptyset \land \forall x \in T(M_w): xx\in T(M_w)\land xxx\notin T(M_w)\}$

I showed that the language $ L’=\{w | T(M_w)\neq \emptyset\}$ is undecidable due to the Rice Theorem. I created to Turing Machines $ M_1$ with $ T(M_1)=\emptyset$ and $ M_2$ with $ T(M_2)=\Sigma^*$ to show that the language L’ is not trivial. (since $ 1\in L$ and $ 2\notin A$ . With out loss of generality i assume those machines have the GĂ¶delindexes 1 and 2)

My problem now is, that I know no way to show, that this result transfers to the language L. I know that the language L must only contain such GĂ¶del Indexes, that for those indexes the following TM must accept infinite words (because in case of $ x\in T(M_w)$ there must be $ xx\in T(M_w)$ … and therefore must be $ xxxx\in T(M_w)$ etc.)

I would love to hear suggestions / answers! Thanks in advance