Can we guarantee that the computation on the Turing machine in our case will never stop?

Question

Assume that we have a Turing machine which on an empty input, with $n > 100$ states worked $n^{ n^n}$ steps. Can we guarantee that in this case the computation will never stop?

Answer 1

I'll elaborate a bit on my comments. Let $f$ be any computable function, for example $f(n)=n^{n^n}$. If an $n$-state Turing machine running for $f(n)$ steps was sufficient to prove that it never halts, then you could solve the halting problem by simply simulating the TM for $f(n)$ steps. Since the halting problem is undecidable, no computable function of the number of states can give an upper bound for how long a TM can run before halting. Incidentally, there is such an upper bound in the form of the busy beaver function, but the catch is that it is not computable.