Condition for Existence of Third Number Proportional to Two Numbers

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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 $18$)


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.


Historical Note

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