functions - How to write down a bijection from $mathbb N$ onto $mathbb Ntimesmathbb Ntimesmathbb N$?

I'm looking for a bijection from $\mathbb N$ to $\mathbb N\times\mathbb N\times\mathbb N$.

• I know it exists


• I know an informal way to define one, ordering the triplets of natural numbers in a 3D array an counting them diagonally as they do here with 2D array.


• And I know a bijection from $\mathbb N\times\mathbb N\times\mathbb N$ to $\mathbb N$ from this answer which uses factorization

But how to write it explicitly?

Thank you very much.

Answer

Use Cantor's pairing function $\langle \cdot, \cdot \rangle: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ twice: $(a,b,c) \mapsto \langle \langle a,b \rangle, c \rangle$. Cantor's pairing function has an inverse which is easily-stated if a bit fiddly (see the proof of bijectivity on the linked Wikipedia page).