Comment by lanstin

Diophantine equations are as they say Turing complete. That is for any question about does this Turing machine with this tape halt with a certain value there is a corresponding Diophantine equation, which has solutions if the machine halts with the output corresponding to the values it is solved by. I think this paper covers it for register machines rather than Turing machines directly: https://carleton.ca/math/wp-content/uploads/Nick-Murphy-Hono...