I'm not an expert in this problem (nor do I ponder number theory that often), but here are my thoughts: Countable doesn't mean finite. It's possible that ther

I'm not an expert in this problem (nor do I ponder number theory that often), but here are my thoughts: Countable doesn't mean finite. It's possible that there is a sum of three cubes with "big" magnitudes that sum to a "small" integer (just see this solution for 42). The tweet chain gives a type of integer that we've proven cannot be expressed as a sum of three cubes (9k+4 and 9k+5), but that only means we've proven anything for 2/9ths of the integers, and it's only conjectured that the rest can be expressed as such. There are some trivial other things can we can prove, such as any number that is 3x a cube, or 2x a cube plus/minus another cube. But that doesn't get us to full coverage. As of yet we haven't proven a result for enough patterns such that every integer belongs to one of those patterns. I suppose to answer your question is that no, we don't have enough small numbers proven and/or we don't have a recursion pattern that covers the integers with the known numbers we have yet. That's just my layman's understanding though. As it stands now, we do know that there are infinite numbers that cannot be expressed as a sum of three cubes, but we do have a useful finite set of integers that can. So a full solution to the problem will be good to know, but we already know that an answer is "some can, some can't." Edit: I realize that someone is going to take issue with my changing GP saying enumerable to me saying countable. There's a difference, and I guess it could matter, but I'm too honest to change my original wording. Sorry in advance for getting it wrong.