There may be years of investigation as to how far you can generalize these methods. As to how central it is, it's a longstanding problem that Erdos loves to cite for that branch of math.

The thing is is that it seems a lot of the effort through the years (which is unquantifiable in scale as to how much time was spent and how many people focused their entire worklives on it if any) has gone for trying to look for the proof, and the search for the disproof seems minimal.