So whole numbers have a unique representation as the product of primes raised to a certain power. This representation can be extended to rational numbers too if we allow negative powers, since if $$a:=\prod_{i}^{N_a} p_i^{a_i}$$ and $$b:=\prod_{j}^{N_b} p_j^{b_j}$$ are coprimes then their ratio $$x:=\frac{a}{b} = \prod_{i}^{N_a} p_i^{a_i}\times\prod_{j}^{N_b} p_j^{-b_j}$$ is uniqely represented by the set of pairs $(a_i, i)$ and $(b_j, j)$. I denote this representation with $R_1(x)$.
Rational numbers have another uniqe representation in the form of finite continued fractions $$x = x_0 + \frac{1}{x_1+\frac{1}{x_2+…}} = [x_0;x_1,x_2…x_N]$$ where each $x_i \in \Bbb{N^+}$. This can be reformulated in a way that $x$ is represented by the set of representations $$R_2(x):=\{(R_1(x_i),i)\,|\,0\le i\le N\}$$ My question basically is how can I go from $R_2$ to $R_1$ without re-calculating prime factorizations of sums arising during the trivial "$R_2$ to $R_1$" process (or is this even possible)? If not, then does $R_2$ contain any "useful" and "easily accessible" information about $R_1$?
Edit: I thought it would be easier to answer the question if you knew my motivation: I'd like to extend the concept of prime factorization further to the set of irrationals. Since we know the $R_2$ of an irrational it would be somehow logical to use this to obtain its $R_1$-like representation, that should fulfill the uniqueness, but can be infinitely long.
Best Answer
I do not believe that continued fractions contain useful information about the prime factorization of the numerator and denominator. Given two large coprime integers $a, b$ of similar magnitude, it is quite fast (using the Euclidean algorithm) to construct the continued fraction of $a/b$, which usually will involve fairly small terms. On the other hand, factoring $a$ and $b$ is notoriously difficult.