Square of Coprime Number is Coprime

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Let $a$ and $b$ be coprime integers:

$a, b \in \Z: a \perp b$


$a^2 \perp b$

In the words of Euclid:

If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

(The Elements: Book $\text{VII}$: Proposition $25$)


Let $a \perp b$.

Let $a^2 = c$.

Let $d = a$.

As $a \perp b$ it follows that $d \perp b$.

From Proposition $24$ of Book $\text{VII} $: Integer Coprime to Factors is Coprime to Whole:

$a d \perp b$

But $a d = c = a^2$.

Hence the result.


Historical Note

This proof is Proposition $25$ of Book $\text{VII}$ of Euclid's The Elements.