Powers of Coprime Numbers are Coprime
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 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 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
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions