# Condition for Existence of Third Number Proportional to Two Numbers

## Theorem

Let $a, b, c \in \Z$ be integers.

Let $\tuple {a, b, c}$ be a geometric sequence.

In order for this to be possible, both of these conditions must be true:

- $(1): \quad a$ and $b$ cannot be coprime
- $(2): \quad a \divides b^2$

where $\divides$ denotes divisibility.

In the words of Euclid:

*Given two numbers, to investigate whether it is possible to find a third proportional to them.*

(*The Elements*: Book $\text{IX}$: Proposition $18$)

## Proof

Let $P = \tuple {a, b, c}$ be a geometric sequence.

Then by definition their common ratio is:

- $\dfrac b a = \dfrac c b$

From Two Coprime Integers have no Third Integer Proportional it cannot be the case that $a$ and $b$ are coprime.

Thus condition $(1)$ is satisfied.

From Form of Geometric Sequence of Integers, $P$ is in the form:

- $\tuple {k p^2, k p q, k q^2}$

from which it can be seen that:

- $k p^2 \divides k^2 p^2 q^2$

demonstrating that condition $(2)$ is satisfied.

$\blacksquare$

## Historical Note

This proof is Proposition $18$ of Book $\text{IX}$ of Euclid's *The Elements*.

## Sources

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