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.



Simplest diophantine equation with open solvability
Determining which integers $n$ are a sum of three cubes is a very famous open problem:


$$a^3 + b^3 + c^3 = n, \quad a,b,c \in \mathbb{Z}.$$
Conjecturally, $n$ is a sum of three cubes iff $n \not \equiv 4,5 \bmod 9$.


Note that this is really a family of Diophantine equations, rather than a single Diophantine equation.


