In my paper I would like to include an example of easy concrete finitary statement which can be easily verified by probabilistic algorithm to any reasonable confidence level, but which looks completely hopeless to prove rigorously.

A candidate example is the statement "$10^{1500} + 2329$ is prime". This can be easily checked with (probabilistic) Miller–Rabin primarity test, while deterministic AKS primality test is too slow to work in this region. However, I am not sure if some other methods (based on elliptic curves, etc.) could be used to rigorously and efficiently prove that this particular number is prime.

Does there exists $n$ such that Miller–Rabin primarity test works efficiently for numbers of the order $10^n$, but all rigorous methods are expected to work "forever"?

Alternatively, can you suggest any other concrete statement which works? For example, based on polynomial identity testing? That is, statement "$P=0$" (or "$P \neq 0$") for a concrete polynomial P, simple enough to be included in the paper explicitly.