# Does there exist a one-to-one function from $\mathbb{N}^5 \mapsto \mathbb{N}^3$

real-analysis

I'm working on a coding project and I have a 5-dimensional vector $$\mathbf{v} \in \mathbb{N}^5$$ that I need to map (in a unique fashion) to RGB colorspace (so, essentially I just want to map $$\mathbf{v}$$ to some $$\mathbf{r} \in$$ $$\mathbb{N}^3$$).

I've been able to find Cantor's Pairing Function, which was quite useful for a different purpose I had but doesn't seem to be of use here. So, my question does there exist such an injection from $$\mathbb{N}^5 \mapsto \mathbb{N}^3$$?

EDIT:In case you're interested, for $$k$$ a natural, $$|\mathbb N^k|= |\mathbb N |$$ Using Schroeder-Bernstein, which states that sets $$A$$ and $$B$$ have the same cardinality iff there is an injection from $$A$$ to $$B$$ and an injection from $$B$$ to $$A$$ . You have an obvious injection in one side (e.g.,$$n \rightarrow (n,0,...,0)$$ ) , and in the opposite direction : Select different primes $$p_1, p_2,..,p_k$$ and map $$(n_1,n_2,..,n_k) \rightarrow p_1^{n_1}p_2^{n_2}..p_k^{n_k}$$