Power of Two is Even-Times Even Only

Theorem

Let $a > 2$ be a power of $2$.

Then $a$ is even-times even only.

In the words of Euclid:

Each of the numbers which are continually doubled beginning from a dyad is even-times even only.

(The Elements: Book $\text{IX}$: Proposition $32$)

Proof

As $a$ is a power of $2$ greater than $2$:

$\exists k \in \Z_{>1}: a = 2^k$

Thus:

$a = 2^2 2^{k - 2}$

and so has $2^2 = 4$ as a divisor.

Let $b$ be an odd number.

By definition:

$b \nmid 2$

The result follows by Integer Coprime to Factors is Coprime to Whole.

$\blacksquare$

Historical Note

This proof is Proposition $32$ 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