Abstract
Let 0 < a < b be two relatively prime integers and let a, b be the numerical semigroup generated by a and b with Frobenius number g(a, b) = ab -a -b. In this note, we prove that there exists a prime number p ∈ a, b with p < g(a, b) when the product ab is sufficiently large. Two related conjectures are posed and discussed as well.