Once you hear about the uncomputability of such problems, it makes you see that all Diophantine equation questions risk being undecidable, though in some simpler cases we manage to come up with answers. The feeling is similar to watching people trying to solve the Halting problem, e.g. in the effort to determine BB(5).
www.jstor.org/stable/1970438 says for prime modulo there is an algorithm.
Question for non-prime modulo: math.stackexchange.com/questions/4944623/are-diophantine-equations-decidable-in-modular-arithmetic
TODO is it or is not:
- mathoverflow.net/questions/207482/algorithmic-un-solvability-of-diophantine-equations-of-given-degree-with-given
- math.stackexchange.com/questions/181380/second-degree-diophantine-equations
- mathoverflow.net/questions/142938/is-there-an-algorithm-to-solve-quadratic-diophantine-equations
- math.stackexchange.com/questions/798609/is-there-any-solution-to-this-quadratic-diophantine-equation
mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/103415#103415 provides a specific single undecidable Diophantine equation.
mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/11557#11557 contains a good overview of the decidability status of variants over rings other than the integers.
