The delta set of a numerical semigroup S, denoted ∆(S), is a factorization invariant that measures the complexity of the sets of lengths of elements in S. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup S? It is known that min ∆(S) = gcd ∆(S) is a necessary condition. For any two-element set {d, td} we produce a semigroup S with this delta set. We then show that for t ≥ 2, the set {d, td} occurs as the delta set of some numerical semigroup of embedding dimension three if and only if t = 2.
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