I have a problem with this task:
Is there a Turing machine $M$ able to write the following language, where a $\langle M \rangle$ is the usual encoding of the machine $M$?
The language is: $L = \{ w \in \{0,1 \}^* | 1 \langle M \rangle w ; \text{is a prime number}\}$ .
Can we use recursion theorem to prove this, please? Or do you have idea how to proceed?