  For any sufficiently strong theory of arithmetic, the set of Diophantine
equations provably unsolvable in the theory is algorithmically undecidable, as
a consequence of the MRDP theorem. In contrast, we show decidability of
Diophantine equations provably unsolvable in Robinson's arithmetic Q. The
argument hinges on an analysis of a particular class of equations, hitherto
unexplored in Diophantine literature. We also axiomatize the universal fragment
of Q in the process.
