[yorwba]

The correspondence is bidirectional. The Diophantine equation you mention encodes a Turing machine that halts iff the Diophantine equation is solvable, and the proof of the independence (impossible to prove true nor false) of the halting problem for that specific Turing machine from Zermelo-Fraenkel set theory implies that whether or not the Diophantine equation in question has solutions is also independent from Zermelo-Fraenkel set theory.