TITLE: How can there be explicit polynomial equations for which the existence of integer solutions is unprovable?
QUESTION [10 upvotes]: This answer suggests that there are explicit polynomial equations for which the existence 
(or nonexistence) of integer solutions is unprovable. How can this be?

REPLY [3 votes]: The answer to this question depends on how the problem is defined, but the answer is no, at least without defining the problem in a misleading way.
Since my first solution was completely off the mark, I have deleted it and posted this new one.
Consider polynomial p. If it has an integer solution, then the solution will eventually be found by random guessing. So if it is impossible to prove the existential status, there must be no solution.
Now we know from this link that there is a polynomial, q, that is unsolvable in the integers iff ZFC is consistent. It is well known that ZFC cannot prove its own consistency. So if it ZFC is consistent, then q is unsolvable, but we cannot prove this as then we could prove ZFC. So it seems like it is accurate to say that if mathematics is consistent, we have a polynomial with no integer roots, but we can't prove it. However, if we are assuming maths is consistent, we can use this to prove that the equation is unsolvable (indeed that is what we have done). So, it really isn't accurate an accurate statement at all. 
To further clarify, when considering mathematical truth, there are two basic ways of viewing it. The first is where we are assuming that are axioms are true, which necessarily means assuming consistency. If we show any problem is equivalent to consistency, then we consider it to be true.
The other is where we are considering a formal set of statements, of which the axioms have been defined to be true and seeing which statements can be derived to be true. From this viewpoint, we don't actually know whether the axioms are consistent or not. In fact, Godel's second incompleteness theorem shows that no "non-trivial" atomic system can prove its own consistency. So showing a problem is equivalent to consistency is actually the same as showing that the problem is unprovable by the atomic system.
The confusion comes from assuming ZFC is consistent to eliminate one possibility in a choice, yet not allowing this assumption to be used as an axiom in the proofs.