A generalization of the Erdős-Birch theorem

Abstract

Let $N_0$ denote the set of all nonnegative integers. A set $T$ of positive integers is called complete if every sufficiently large integer is the sum of distinct integers taken from $T$. In 1959, Birch confirmed a conjecture of Erdős by proving that the set $\{s_1^{x_1}{s}_2^{x_2}:x_1,x_2\in N_0\}$ is complete if $s_1,s_2>1$ are integers with $gcd(s_1,s_2)=1$. In this paper, the following result is proved: if $s_1,\dots ,s_k>1$ are integers, then the set $\{s_1^{x_1}\cdots s_k^{x_k}:x_1,\dots ,x_k\in N_0\}$is complete if and only if $gcd(s_1,\dots ,s_k)\le 2$.