Is it worthwhile to try to prove a conjecture by mapping it to a Turing machine? 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.
First, you don't need to turn them into Turing machines, it is essentially the same as running a proof search algorithm. The logical complexity of the formula (which in the cases you have in mind are $\Pi^0_1$) has no effect on this, searching for proofs in any effective theory (like ZFC, for proof to make sense you have to fix a theory) can be done in a straight forward manner, the set of provable statements for any effective theory is $\Sigma^0_1$ and therefore computable enumerable.  But there are no reasons to do that, you don't gain anything a priori just by adding one extra unnecessary layer of coding, it has no benefit (AFAIK).

Also don't confuse proof search with a direct reduction from the truth of a $\Pi^0_1$ statement to the halting problem (which exists as not halting is $\Pi^0_1$-complete).

Second, consider the much simpler case of propositional tautologies in place of first-order statements like GC. (propositional tautologies is a coNP set also known as $\Pi^B_1$). The state of art proof search algorithms (i.e. SAT solvers) are exponentially inefficient in finding proofs of very simple finitary mathematical statements like the pigeon-hole principle tautologies. You can run a first-order proof search program to look for a ZFC proof of GC but I haven't seen any interesting mathematical statement being discovered/proven by unguided brute-force search of proofs. So there is no example of this being useful. (For human-guided use of computer in proving mathematical theorems we have the four colors theorem.)

Third, if you use a proof search algorithm you will only know if there is a proof, if there isn't any you will not know and the algorithm will continue forever without halting. In other words, you can't write a program like the one you have stated in the question (which is why I commented that this is more suitable for CS Q&amp;A site since this is very basic mistake in computability theory).

See also my answers to similar questions on math and CS.