Elements of Geometric Sequence from One which are Powers of Number

Theorem
Let $G_n = \left\langle{a_n}\right\rangle_{1 \mathop \le i \mathop \le n}$ be a geometric progression of integers.

Let $a_1 = 1$.

Then:
 * $\forall m, k \in \left\{{1, 2, \ldots, n}\right\}: k \mathop \backslash m \implies a_{m-1}$ is a power of $k$

where $\backslash$ denotes divisibility.

Proof
By Form of Geometric Progression of Integers, the general term of $G_n$ can be expressed as:
 * $a_j = k q^{j - 1} p^{n - j}$

for some $k, p, q \in \Z$.

As $a_1 = 1$ it follows that $k = 1$ and $p = 1$.

Thus:
 * $a_j = q^{j - 1}$

for some $q \in \Z$.

Let $k, m \in \left\{{1, 2, \ldots, n}\right\}$ such that $k \mathop \backslash m$.

By definition of divisibility:
 * $\exists r \in \Z: m = r k$

Then:

That is, $a_{m - 1}$ is a power of $k$.