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