Two Coprime Integers have no Third Integer Proportional

Theorem

Let $a, b \in \Z_{>0}$ be integers such that $a$ and $b$ are coprime.

Then there is no integer $c \in \Z$ such that:

$\dfrac a b = \dfrac b c$

In the words of Euclid:

If two numbers be prime to each other, the second will not be to any other number as the first is to the second.

(The Elements: Book $\text{IX}$: Proposition $16$)

Proof

Suppose such a $c$ exists.

From Coprime Numbers form Fraction in Lowest Terms, $\dfrac a b$ is in canonical form.

From Ratios of Fractions in Lowest Terms:

$a \divides b$

where $\divides$ denotes divisibility.

This contradicts the fact that $a$ and $b$ are coprime.

Hence such a $c$ cannot exist.

$\blacksquare$

Historical Note

This proof is Proposition $16$ 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