Elements of Geometric Sequence from One which Divide Later Elements

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

Let $a_0 = 1$.

Let $m \in \Z_{> 0}$.


$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$

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 less measures the greater according to some one of the numbers which have place among the proportional numbers.

(The Elements: Book $\text{IX}$: Proposition $11$)


In the words of Euclid:

And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.

(The Elements: Book $\text{IX}$: Proposition $11$ : Porism)


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 $q \in \Z$.

Hence the result from Divisors of Power of Prime.


Historical Note

This proof is Proposition $11$ of Book $\text{IX}$ of Euclid's The Elements.