ABSTRACTWe 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∈ℤ+ with n≥3 and 4≤m≤n3 there is an atomic domain with precisely n atoms, precisely m maximal ideals and no prime elements. The proofs involve an interplay of commutative algebra, algebraic number theory and additive number theory.