Definition:Geometric Sequence of Integers in Lowest Terms

Definition
Let $G_n = \left\langle{a_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ be a geometric progression of integers.

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

Let $S$ be the set of all such geometric progressions:
 * $S = \left\{{G: G}\right.$ is a geometric progression 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 $S$:


 * $\forall Q = \left\langle{b_j}\right\rangle_{0 \mathop \le j \mathop \le n} \in S: \forall j \in \left\{{0, 1, \ldots, n}\right\}: \left\vert{a_j}\right\vert \le \left\vert{b_j}\right\vert$

Also see

 * Form of Geometric Progression of Integers in Lowest Terms
 * Geometric Progression in Lowest Terms has Coprime Extremes
 * Geometric Progression with Coprime Extremes is in Lowest Terms
 * Construction of Geometric Progression in Lowest Terms