# How do I prove that SPACE($n^{555}$) $\neq$ NP?

I thought about finding a language with a polynomial verificator "larger" than $$n^{555}$$, but then I realized it would not imply the space needed for computation is the size of the verificator.

According to the space hierarchy theorem, there is a language in $$\mathrm{SPACE}(n^{556})$$ which is not in $$\mathsf{SPACE}(n^{555})$$. A padded version of this language will be in $$\mathsf{SPACE}(n^{555})$$. If the latter is equal to $$\mathsf{NP}$$, then the original language would also be in $$\mathsf{NP} = \mathsf{SPACE}(n^{555})$$, since we can implement padding while remaining inside $$\mathsf{NP}$$.