  In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$,
there is a straightforward procedure to determine if an element is prime by
simply reducing to a direct check of finitely many potential divisors. Despite
the fact that such a naive approach does not immediately translate to integral
domains like $\mathbb{Z}[x]$ or the ring of integers in an algebraic number
field, there still exist computational procedures that work to determine the
prime elements in these cases. In contrast, we will show how to computably
extend $\mathbb{Z}$ in such a way that we can control the ordinary integer
primes in any $\Pi_2^0$ way, all while maintaining unique factorization. As a
corollary, we establish the existence of a computable UFD such that the set of
primes is $\Pi_2^0$-complete in every computable presentation.
