The Frobenius problem for numerical semigroups with embedding dimension equal to three

If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = 〈a, b, cb − da〉 with a, b, c, d positive integers such that gcd(a, b) = gcd(a, c) = gcd(b, d) = 1, c ∈ {2, . . . , a− 1}, and a < b < cb− da. In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseudo-Frobenius numbers of 〈a, b, cb− da〉 in the case in which the interval [ a c , b d ] contains some integer.