Formulaic representation of the lowest form of the denominator of a rational number

Question

Given an arbitrary rational number $p$, is there a way to find out the denominator of its simplest form? In other words, can we formulaically (non-algorithmically) find the lowest integer denominator of all the ways $p$ can be represented?

Suppose $p$ = $\frac{n}{d}$ is the lowest form. Then, I am looking for an $f(x)$ such that $f(p)$ would have produced $d$. As an extension, can there also be a $g$ such that $g(p)$ would have been $n$?

I played around with inverses, proportions, and integer division, but I can't seem to figure out [1] if this can be done and if so, [2] how.

Answer 1

There is no formula of the type that I think you want. There is a good algorithm. Namely, you let $a_0=[p]$, let $b_0=(p-a_0)^{-1}$, $a_i=[b_{i-1}]$, $b_i=(b_{i-1}-a_i)^{-1}$, continuing until $b_i$ blows up. You then have the continued fraction partial quotients $a_0,a_1,\dots,a_n$ for $p$, from which you reconstruct $p$ in lowest terms.

Answer 2

The short answer is "no" because the whole point of rationals is that they're equal regardless of common factors and there's no way for anything to distinguish between them on purpose because that's just what we mean by "equal". Any means of distinguishing them would have to be an "algorithm" which operates on their parts and not a "formula" which operates on the numbers.