Can a Turing Machine decide if an NFA accepts a string of prime length? I want to know if the following problem is decidable:

Instance: An NFA A with n states

Question: Does there exist some prime number p such that A accepts some string of length p.

My belief is that this problem is undecidable, but I can't prove it. The decider can easily have an algorithm to figure out if a particular number is prime, but I don't see how it would be able to analyze the NFA in enough detail to know exactly what lengths it can produce. It could start testing strings with the NFA, but for an infinite language, it may never halt (and thus not be a decider).

The NFA can easily be changed to a DFA or regular expression if the solution needs it, of course.

This question is something I've been pondering as a self-made prep question for a final I have coming up in 2 weeks.
The lengths of the strings accepted by a DFA form a semilinear set (like in Parikh's theorem for context free languages), the description of those isn't too hard to come by (essentially splice up all possible cycles of the automaton), and by Dirichlet's theorem any arithmetic progression of the form $a + b k$ with $\gcd(a, b) = 1$ contains an infinitude of primes.

Pulling the above together gives an algorithm to check if your regular (or even context free language) contains strings of prime length. Definitely not a simple question, IMVHO...