Definition:Geometric Sequence of Integers in Lowest Terms

Definition

Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers.

Let $r$ be the common ratio of $G_n$.

Let $S$ be the set of all such geometric sequence:

$S = \left\{{G: G}\right.$ is a geometric sequence of integers whose common ratio is $\left.{r}\right\}$

Then $G_n$ is in lowest terms if the absolute values of the terms of $G_n$ are the smallest, term for term, of all the elements of $S$:

$\forall Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n} \in S: \forall j \in \set {0, 1, \ldots, n}: \size {a_j} \le \size {b_j}$