What is the simplest diophantine equation for which we (collectively) don't know whether it has *any* solutions? I'm aware of many simple ones where we don't know (whether we know) *all* the solutions, but all of these that I know have some solution.

Yes, I know that "simplest" is subjective. I'd be satisfied if it could be typeset in one line in a LaTeX document. Also, it would be nice if it could be easily memorized (no seven-digit numbers with nonobvious patterns:) though that's secondary.