Elements of Geometric Sequence between Coprime Numbers

Theorem

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n + 1$.

Let $a_0$ be coprime to $a_n$.

Then there exist geometric sequences of integers $Q_1$ and $Q_2$ of length $n + 1$ such that:

the initial term of both $Q_1$ and $Q_2$ is $1$

the final term of $Q_1$ is $a_0$

the final term of $Q_2$ is $a_n$.

In the words of Euclid:

If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

(The Elements: Book $\text{VIII}$: Proposition $9$)

Proof

Let the common ratio of $P$ be $r$.

By Form of Geometric Sequence of Integers with Coprime Extremes, the $j$th term of $P$ is given by:

$a_j = q^j p^{n - j}$

such that:

$a_0 = p^n$

$a_n = q^n$

For $j \in \set {0, 1, \ldots, n}$, let $r_j = q^j$.

Let the finite sequence $Q_2 = \sequence {t_j}_{0 \mathop \le j \mathop \le n}$ be defined as:

$\forall j \in \set {0, 1, \ldots, n}: t_j = p^{n - j}$

Then $Q_1$ is a geometric sequence of integers of length $n$ such that:

$t_0 = 1$

$t_n = p^n$

Let the finite sequence $Q_1 = \sequence {s_j}_{0 \mathop \le j \mathop \le n}$ be defined as:

$\forall j \in \set {0, 1, \ldots, n}: s_j = q^j$

Then $Q_1$ is a geometric sequence of integers of length $n$ such that:

$s_0 = 1$

$s_n = q^n$

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $9$ of Book $\text{VIII}$ of Euclid's The Elements.
It is the converse of Proposition $10$ of Book $\text{VIII} $: Product of Geometric Sequences from One.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions