As one might know, the Stern Brocot tree elegantly and compactly models all rational numbers.
I am now left wondering if a process like this tree modeling could be done not only for pairs but for triples and even beyond. To be more specific, what minimal operations does one need if this can be done simply, can it be done only having triples and then working from there with only some version of addition between elements?
There’s a simple bijection between pairs of coprime pairs and triples of coprime numbers. For any pair of coprime pairs $a,b$ and $c,d$, note that $ac,bc,d$ is a coprime triple. For any triple $x,y,z$ of coprime numbers, let $c$ be the gcd of $x$ and $y$, then $x/c,y/c$ are coprime as well as $c,z$. These two operations are inverses thereby establishing a bijection.
Thus, you can take your favorite characterization of coprime numbers (i.e. Stern Brocot), and use it to describe coprime triplets (or tuples of any length).
