Primes in numerical semigroups

Abstract

Let $0 be two relatively prime integers and let $\langle a,b\rangle $ 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\in \langle a,b\rangle $ with $p when the product $ab$ is sufficiently large. Two related conjectures are posed and discussed as well.