Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd

Theorem

Let $a \in \Z$ be an integer such that:

$(1): \quad a$ is not a power of $2$

$(2): \quad \dfrac a 2$ is an even integer.

Then $a$ is both even-times even and even-times odd.

In the words of Euclid:

If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both even-times even and even-times odd.

(The Elements: Book $\text{IX}$: Proposition $34$)

Proof

As $\dfrac a 2$ is an even integer it follows that:

$\dfrac a 2 = 2 r$

for some $r \in \Z$.

That is:

$a = 2^2 r$

and so $a$ is even-times even by definition.

$\Box$

Aiming for a contradiction, suppose $a$ is not even-times odd.

Then $a$ does not have an odd divisor.

Thus in its prime decomposition there are no odd primes.

Thus $a$ is in the form:

$a = 2^k$

for some $k \in \Z_{>0}$.

That is, $a$ is a power of $2$.

This contradicts condition $(1)$.

From this contradiction it is deduced that $a$ is even-times odd.

$\blacksquare$

Historical Note

This proof is Proposition $34$ 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