Powers of Coprime Numbers are Coprime

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Theorem

Let $a, b$ be coprime integers:

$a \perp b$


Then:

$\forall n \in \N_{>0}: a^n \perp b^n$


In the words of Euclid:

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].

(The Elements: Book $\text{VII}$: Proposition $27$)


Proof

Proof by induction:

Let $a \perp b$.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:

$a^n \perp b^n$


$\map P 1$ is true, as this just says:

$a \perp b$


Basis for the Induction

By Proposition $25$ of Book $\text{VII} $: Square of Coprime Number is Coprime:

$a^2 \perp b$

Again, by Proposition $25$ of Book $\text{VII} $: Square of Coprime Number is Coprime:

$a^2 \perp b^2$

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$a^k \perp b^k$


Then we need to show:

$a^{k + 1} \perp b^{k + 1}$


Induction Step

This is our induction step:


\(\ds a^k\) \(\perp\) \(\ds b^k\)
\(\ds \leadsto \ \ \) \(\ds a^k \times a\) \(\perp\) \(\ds b^k\) Proposition $24$ of Book $\text{VII} $: Integer Coprime to all Factors is Coprime to Whole
\(\ds \leadsto \ \ \) \(\ds a^{k + 1}\) \(\perp\) \(\ds b^k\)
\(\ds \leadsto \ \ \) \(\ds a^{k + 1}\) \(\perp\) \(\ds b^k \times b\) Proposition $24$ of Book $\text{VII} $: Integer Coprime to all Factors is Coprime to Whole
\(\ds \leadsto \ \ \) \(\ds a^{k + 1}\) \(\perp\) \(\ds b^{k + 1}\)

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \N_{>0}: a^n \perp b^n$

$\blacksquare$


Historical Note

This proof is Proposition $27$ of Book $\text{VII}$ of Euclid's The Elements.
Euclid's proof does not use the full induction process, which is a later mathematical innovation.


Sources