Construction of Geometric Sequence in Lowest Terms/Proof 1

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Theorem

It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms.


In the words of Euclid:

To find numbers in continued proportion, as many as may be prescribed, and the least of those that are in a given ratio.

(The Elements: Book $\text{VIII}$: Proposition $2$)


Proof

Let $r = \dfrac a b$ be the given common ratio.

Let the required geometric sequence have a length of $4$.

Let $a^2 = c$.

Let $a b = d$.

Let $b^2 = e$.

Let:

$a c = f$
$a d = g$
$a e = h$

and let:

$b e = k$

As:

$a^2 = c$
$a b = d$

it follows from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers that:

$\dfrac a b = \dfrac c d$

As:

$a b = d$
$b^2 = e$

it follows from Proposition $18$ of Book $\text{VII} $: Ratios of Multiples of Numbers that:

$\dfrac a b = \dfrac d e$

As:

$a c = f$
$a d = g$

it follows from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers that:

$\dfrac c d = \dfrac f g$

As:

$a d = g$
$a e = h$

it follows from Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers that:

$\dfrac d e = \dfrac g h$

As:

$a e = h$
$b e = k$

it follows from Proposition $18$ of Book $\text{VII} $: Ratios of Multiples of Numbers that:

$\dfrac a b = \dfrac h k$


Putting the above together:

$c, d, e$ are in geometric sequence with common ratio $\dfrac a b$
$f, g, h, k$ are in geometric sequence with common ratio $\dfrac a b$


We have that $a$ and $b$ are the smallest numbers with the same ratio.

So by Proposition $22$ of Book $\text{VII} $: Numbers forming Fraction in Lowest Terms are Coprime:

$a \perp b$

where $\perp$ denotes coprimality.


We also have that:

$a^2 = c, b^2 = e$

and:

$a c = e, b e = k$

so by Proposition $27$ of Book $\text{VII} $: Powers of Coprime Numbers are Coprime:

$c \perp e$
$f \perp k$

But from Proposition $1$ of Book $\text{VIII} $: Geometric Sequence with Coprime Extremes is in Lowest Terms, these are the least of those with the same common ratio.

Hence the result.

$\blacksquare$


Historical Note

This proof is Proposition $2$ of Book $\text{VIII}$ of Euclid's The Elements.
This proof as given by Euclid takes the special case of four terms and expects the reader to extrapolate from there.


Sources