  We study the number of atoms and maximal ideals in an atomic domain with
finitely many atoms and no prime elements. We show in particular that for all
$m,n \in \mathbb{Z}^+$ with $n \geq 3$ and $4 \leq m \leq \frac{n}{3}$ there is
an atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and no
prime elements. The proofs use both commutative algebra and additive number
theory.
