Square of Coprime Number is Coprime
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Theorem
Let $a$ and $b$ be coprime integers:
- $a, b \in \Z: a \perp b$
Then:
- $a^2 \perp b$
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 $25$)
Proof
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.
$\blacksquare$
Historical Note
This proof is Proposition $25$ of Book $\text{VII}$ 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 $\text{VII}$. Propositions