Divisibility properties of additive bases

Abstract:

Let $h \geqslant 2$. There exists an asymptotic basis $A$ of order $h$ such that $({a_1}, \ldots ,{a_k}){\text { > 1}}$ for all ${a_1} \ldots ,{a_k} \in A$ if and only if $k{\text { < }}h$. If $k \geqslant h$, the sumset $hA$ contains only composite numbers. For $h = k$, there exists a set $A$ of nonnegative integers with $({a_1}, \ldots ,{a_h}){\text { > }}1$ for all ${a_1}, \ldots ,{a_h} \in A$ such that for every prime $p$ the sumset $hA$ contains all sufficiently large multiples of $p$.