Geometric Sequence with Coprime Extremes is in Lowest Terms

Theorem
Let $G_n = \left\langle{a_1, a_2, \ldots, a_n}\right\rangle$ be a geometric progression where $a_1, \ldots, a_n$ are all natural numbers.

Let:
 * $a_1 \perp a_n$

where $\perp$ denotes coprimality.

Then the terms of $G_n$ are the lowest possible natural numbers with the same common ratio.

Proof
Let $G_n = \left\langle{a_1, a_2, \ldots, a_n}\right\rangle$ be natural numbers in geometric progression such that $a_1 \perp a_n$.

Let $G\,'_n = \left\langle{b_1, b_2, \cdots, b_n }\right\rangle$ be another set of natural numbers in geometric progression with the same common ratio where:


 * $\forall k \in \N_{\le n}: a_k > b_k$
 * $\forall k \in \N_{\le n}: a_k : b_k = a_1 : b_1$

From :
 * $a_1 : a_n = b_1 : b_n$

But by hypothesis:
 * $a_1 \perp a_n$

and so from:

and:

it follows that:
 * $a_1 \mathop \backslash b_1$

However, this contradicts the assumption that $b_1 < a_1$.

Therefore $a_1, a_2, \cdots, a_n$ are the least of those with the same common ratio.