Note on a conjecture of Ramírez Alfonsín and Skałba

Let $2<a<b$ be two relatively prime integers and $g=ab-a-b$. It is proved that there exists at least one prime $p\le g$ of the form $p=ax+by(x,y\in {\mathbb{Z}}_{\ge 0})$, which confirms a 2020 conjecture of Ramírez Alfonsín and Skałba.