Prove that $\{0^n 1^{n\cdot m} : n,m \in \mathbb{N}\}$ is not context-free

Question

This is a homework problem I have spent several hours on. A "hint" is given that we may use this fact: If $n,j,k \in \mathbb{N}$ satisfy $ n \geq 2$ and $1 \leq j+k \leq n$, then $n^2+j$ does not evenly divide $n^3+k$.

I cannot find any way to apply this fact. It leads me to believe I should use the string $0^{p^2}1^{p^3}$ or something like that, but I am really just not sure. The pumping lemma has given me trouble since the non regular language version.

Even small hints greatly appreciated at this point.

Answer 1

If you use the pumping lemma on the word $w=0^{p^2} 1^{p^3}$, consider the partition $w=xyzuv$, where $|yzu|\le n$ and $|yu|>0$ ($n$ being the length of $w$). It is easy to prove that from all the cases (that is, from all the possibilities for $y$ and $u$), the only non-trivial case is when $y=0^i$ and $u=1^j$, in which case the hint you mentioned finishes the job.