Is the language $L=\{<M>|M \text{ has 24 states}\}$ recursive?

Question

I know that I cannot use the rice theorem because there are two turing machines that calculate the same function but have different ammount of states. What techniqu can I use to prove that this language is not recursive.

Answer 1

I will assume that by $<M>$ you mean "an encoding of the machine $M$ as an integer".

It depends on your encoding of Turing Machines. We generally suppose that this encoding is at least computable, so that we can actually use Universal Turing Machines, and so on. Hence, you should be able to retrieve the number of states of $M$ given its encoding. Therefore, it is clear that the language $L$ is recursive, and you recognize it using the machine that "decodes" its input $<M>$ and checks whether the number of states of $M$ is equal to 24.