While thinking about primes I just noticed a nice structural relationship between prime-factorization and infinite-dimensional integer vectors and I'm wondering what's known about this.
Here is the setup:
Let $\mathbb Z_{>0}$ be the set of positive integers and $\mathcal P = \{2, 3, 5, \dots\}$ the set of primes. Any positive integer $n$ can uniquely be written as $$n = 2^{k_{n,2}}3^{k_{n,3}}\dots = \prod_{p \in \mathcal P}p^{k_{n,p}},$$ where only finitely many of the coefficients $k_{n,p}\in\mathbb Z_{\ge 0}$ are nonzero. Actually, if we allow the coefficients to take on negative values then we can write each positive rational number $q \in \mathbb Q_{>0}$ in a similar way: $$q = 2^{k_{q,2}}3^{k_{q,3}}\dots = \prod_{p \in \mathcal P}p^{k_{q,p}},$$ for some $k_{q,p} \in \mathbb Z$ (if we assume that the fraction $q$ is fully simplified, the coefficients are uniquely determined).
So we can associate each positive rational number $q$ uniquely with some infinite dimensional vector $$(k_{q,2}, k_{q,3}, \dots).$$ Let's call such a vector a ``prime factorization vector". Note that adding two such vectors corresponds to multiplying their associated positive rational numbers and subtracting them corresponds to dividing their associated positive rational numbers.
Of course, vectors can not only be added and subtracted but one can perform all kinds of geometric operations with them.
Question: What is known about the relationship between the geometry of the ``prime factorization vectors" introduced above and their associated positive rational numbers? (or about the relationship of prime factorization vectors and their associated positive integers?)
