Necessary and sufficient conditions for simple A-bases (Q1065861)

Let A be a set of m distinct integers with \(0\in A\) and \(m\geq 2\). The integral sequence \(B=\{b_ i\}_{i\geq 1}\) is called an A-base for the set of integers provided every integer can be represented uniquely in the form \(n=\sum^{r(n)}_{i=1}a_ i b_ i\), \(a_ i\in A\) for all i. If (with possible rearrangement) B can be written in the form \(B=\{d_ i m^{i-1}\}_{i\geq 1}\) where the \(d_ i\) are integers, then B is called a simple A-base. In this paper we show that A, as above, has a simple A- base if and only if A is a complete residue system modulo m and the elements of A are relatively prime.