David Gale asks:
Query. Is there an easily described sequence which has been PROVED to be finite but we don't know what the last term is?
There used to be: There's a known finite list of principal ideal domains of the form Z[(1+sqrt(1-4k))/2] with k a positive integer, but for a while the state of affairs was that, although the list of known d's had not been proved to be complete, it had been proved that there was at most one other d missing from the list of known examples. So, at *that* time, this would have been an example of what David's looking for; but (a) it no longer is (since the original list has now been shown to be complete), and (b) the list is not so easily described (unless this d, like the number 41, would also have necessarily had the property that n^2 + n + d is prime for all n between 0 and d-2; I'm not sure about this). If you're willing to relax the notion of a sequence and settle for a set, then a contemporary answer to David's question is the set of non-trivial solutions to the Diophantine equation m^p - n^q = 1 (Catalan's equation). It's known that there are at most finitely many. Jim Propp