Landau's fourth problem, which asks whether $x^2+1$ is prime infinitely many times, is almost certainly true, but nobody has been able to prove it for hundreds of years now. It crossed my mind that perhaps this is because it's an undecidable problem, but I think we can know definitely that that's not the case.
Matiyasevich's theorem shows that Diophantine equations can essentially be treated as a Turing-complete system, and as such, is subject is to the halting problem and related pitfalls. My understanding is that this only applies to equations of a certain complexity, and that our current best lower bound is that there are Diophantine equations with $36$ unknowns of at most degree $4$ which are unsolvable.
I can't point to a source, but my own impression was that problems with unknowns of degree $4$ or higher were generally especially difficult (or impossible), while those of degree $2$ were generally readily tractable, with degree $3$ hard to pin down one way or the other. Of course, difficulty is also highly dependent on the number of unknowns.
So with that preamble said, perhaps the simplest formulation I've been able to come up with for Landau's problem as a Diophantine equation is
$$a^2+b^2-c^2=1,\qquad a,b,c\in\mathbb N, \quad a \neq b \neq c.$$
Any solution to this yields a composite $c^2+1$, so if there can be shown that there are arbitrarily large values of $c$ which have no solution, that's a proof, since $c^2+1$ will be prime or twice a prime. There are other ways to state it, but they all have more or less the same structure of a Diophantine equation in three unknowns of degree two.
In general, homogeneous Diophantine equations of this complexity are well understood and typically solvable one way or the other. This one in particular is obviously resistant, but my question is:
Given the general form of the problem, three unknowns of degree two, is that sufficient to know for certain that a solution one way or the other to the problem exists and is provable?
Secondary to this, I am also curious whether or not a viable approach to a proof would be to establish bounds on the number of solutions to a Diophantine equation based on its parameters, if the number of solutions is finite. Such bounds must exist, and even if they were enormous, it might be enough, since solutions to this equation are so abundant. I have tried to look up work along these lines with Diophantine bounds, but haven't found anything I can understand that applies.