Elements of Geometric Sequence from One which are Powers of Number

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

Let $a_0 = 1$.


$\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a power of $k$

where $\divides$ denotes divisibility.

In the words of Euclid:

If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.

(The Elements: Book $\text{IX}$: Proposition $8$)


By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:

$a_j = q^j$

for some $q \in \Z$.

Let $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.

By definition of divisibility:

$\exists r \in \Z: m = r k$


\(\ds a_m\) \(=\) \(\ds q^m\)
\(\ds \) \(=\) \(\ds q^{r k}\)
\(\ds \) \(=\) \(\ds \paren {q^r}^k\)

That is, $a_m$ is a power of $k$.


Historical Note

This proof is Proposition $8$ of Book $\text{IX}$ of Euclid's The Elements.