Lets assume the proof of a conjecture, for example, the famous Goldbach conjecture. Is it possible to try to prove or disprove such a conjecture by devising a Turing machine that accepts if the proof is possible. So, for Turing machine M,
If, the conjecture is provable, accept, else reject.
So, in essence, we don't try to find a proof of the conjecture, but, try to find whether its provable, something like zero-knowledge proofs.
Additional Details: I felt the way most of these conjectures go, is they say: For all n, C holds true. So, in essence, even if it can be shown that C holds up to some n, but, it hasn't been able to be derived for all n. And, this seems to bear a striking similarity to the Halting problem. So, I felt a mapping must be worthwhile to study/research.