Recall that Mills' constant θ is defined such that the floor of θ raised to the power of any positive integer n is a prime number. Start by understanding what this implies about the distribution of primes and how θ might be constructed.

Consider the Prime Number Theorem and how it describes the asymptotic distribution of prime numbers. How might this theorem influence the existence and properties of Mills' constant θ?

To rigorously prove Mills' theorem, you would need to show that there exists a real number θ such that ⌊θ^(3^n)⌋ is prime for all positive integers n. Begin by assuming such a θ exists and explore the recursive relationship between consecutive primes generated by this formula. How does this relate to the gaps between primes?