Not sure if this is the best place to ask this question but I tried to search online, wasn't able to find anything useful.
Consider binary numbers of the form $$x=0|y|0|y$$ where $a|b$ means concatenate the bits of $a$ with the bits of $b$, $y$ is any sequence of bits of size $n$.
Is it possible to find three binary numbers $s_1,s_2,s\neq 0$ of the previous form such that $s_1.s_2=s$ where $.$ is just the plain multiplication operator? If not why?
Obviously, all of $s_1,s_2,s$ need to be of size $\leq 2n+2$ bits
Thanks!
Observe that $x = (2^{n}+1)y$. If you consider $2^n+1$ to be of the desired form, then we're done. Otherwise, this factorization places some constraints on your desired $s$.