The fact that they are fast is irrelevant. Deeper Blue beat Kasparov,
If you want a better example, look at the problems with a Go program.
Alas, I hardly know anything about chess, but indeed I've heard that go is much more based on the typical ability of the human brain to discern patterns and that thus it is much more difficult to design good patterns for it...
Really? If there was a way to calculate in O(1) time the next prime number larger than a given N (which is, essentially, what the OP was asking for), then cryptography that is based on large number factorization is no longer secure. Think about it for a second - it's not that there is a function P(x) that gives you the next prime number, but the work that lead up to it and that will be based on it.
I'm not really sure about that, i.e. that an algorithmically fast primality test (or a function like the one requested by the OP) would imply a fast factorization algorithm. But then I'm far from being an expert in the field and never claimed to be one...
We can go into greater detail offline, if you want.
I must say that while I find all this to be very interesting I'm now dedicating most of my resources to my thesis work (after having lost quite a few years!
) and while it touches occasionally and en passant on number theoretic arguments, basically it has nothing to do with the Theory of Numbers per se.