User:Ascii/Coprime Relation for Integers is Not Reflexive
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Theorem
The relation "is coprime to" on the integers is not reflexive.
That is:
- $\exists n \in \Z: \neg n \perp n$
where $\perp$ denotes "is coprime to".
Proof
Consider $2$.
From Integer Divides Itself, $2$ is a divisor of $2$.
From Absolute Value of Integer is not less than Divisors, $2$ is the greatest divisor of $2$.
So, the greatest common divisor of $2$ and itself is $2$: $\gcd \set {2, 2} = 2$
Hence $\neg \, 2 \perp 2$ and $\perp$ is not reflexive.
$\blacksquare$