Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer."
Some questions (e.g. the existence of a projective plane of order 12) naturally resolve after a finite computation but not feasibly.
I'd like examples of reasonably important open problems that have now been reduced, via nontrivial arguments, to finite but infeasible computations.
I'm sure that additive number theory gives examples (certain questions along the lines of Goldbach conjecture and Waring's problem, but I don't have the details handy). I'd love especially to see examples that don't seem to originate in discrete mathematics.
Best Answer
Computing homotopy groups of spheres has been reduced in several different ways down to a finite but infeasible computation. This was discussed in another thread. John Klein's answer describes an algorithm Dan Kan came up with. The accepted answer points to other work which contains a more efficient method, but which I haven't read. I suppose you could argue that this is not an important enough problem (actually, this has also been done on MathOverflow), but most topologists would disagree. Certainly this is not a problem which originates in discrete math.