Prove that a counterexample exists without knowing one I strive to find a statement $S(n)$ with $n \in N$ that can be proven to be not generally true despite the fact that noone knows a counterexample, i.e. it holds true for all $n$ ever tested so far. Any help?

Define $k$ to be 42 if the Riemann Hypothesis is true, and 108 if it is false.
Now consider $S(n) \equiv n\ne k$.

Alternatively consider $S(n)$ to be "There is a two-symbol Turing machine with 100 states that runs for at least $n$ steps when started on an empty tape, but eventually terminates".
