Let $X$ be an infinite set, i.e., there is an injection $\mathbb N \hookrightarrow X$. There is an obvious injection $X\hookrightarrow X\times X$ (just take $a\in X$ and send any $x \in X$ to $(x,a)$). Is it possible to find (explicitly) an injection $X \times X \hookrightarrow X$?

Don't know if this is helpful: $f: (m,n) \mapsto 2^m(2n+1)-1$ is a bijection from $\mathbb N \times \mathbb N$ to $\mathbb N$.