Numerical semigroups, cyclotomic polynomials and Bernoulli numbers

Abstract: We give two proofs of a folkore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences. In particular, we give a more conceptual reproof of a result of Hong et al. (2012) on gaps between the exponents of non-zero monomials in a binary cyclotomic polynomial. The intent of the author with this expositional paper is to better unify the various results within the cyclotomic polynomial and numerical semigroup communities.