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We study the number of atoms and maximal ideals in an atomic domain withfinitely 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 isan atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and noprime elements. The proofs use both commutative algebra and additive numbertheory.

The number of atoms in an atomic domain