Wait, are you saying they can't be solved by a Turing machine, but they can be solved some other way? If so, how?

The Matiyasevich theorem (which settled Hilbert's 10th Problem) proved that Diophantine equations are fundamentally tied to the Halting Problem. Because a universal Turing machine cannot determine if any arbitrary program will halt, no computer program or mathematical algorithm can ever be written to determine if any arbitrary Diophantine equation has an integer solution.