Show $\{1^n0^m |\space n \neq 2^m\}$ not regular using pumping lemma
Let $p$ be the pumping length, and consider the string $u=1^{2^{p!+p}}$, $w = 0^p$, $v = \epsilon$. Notice $uwv \in L$. According to the pumping lemma, there is a value $q \in \{1,\ldots,p\}$ such that $1^{2^{p!+p}} 0^{p-q+iq} \in L$ for every $i \geq 0$. Choosing $i = 1 + p!/q$, we obtain a contradiction.