A Complete Characterization of Realizable (Embedding Dimension, Multiplicity) Pairs for Complete Intersection Numerical Semigroups

We give a complete characterization of the positive integer pairs $(e,m)$ that occur as the (embedding dimension, multiplicity) of a complete intersection numerical semigroup, thereby settling an open problem in the theory of relations of numerical semigroups. We prove that there exists a complete intersection numerical semigroup $\mathrm{\Gamma }$ with $e(\mathrm{\Gamma })=e$ and $m(\mathrm{\Gamma })=m$ if and only if $(e,m)=(1,1)$ or $e\ge 2$ and $m\ge 2^{e-1}$.