Can the existence of some positive integers satisfying some equations ever be independent of ZFC?

Question

In a subject like number theory there are often open questions along the lines of 'Are there positive integers $a_1 , \ldots , a_n $ satisfying the following (finite number of) equations'. For example Fermat's last theorem for a fixed exponent is such a statement. I was wondering if such a question could ever be independent from ZFC, meaning there is no proof for or against the statement.
I know that the incompleteness theorem states that in our language of arithmetic there has to be some statement that is independent of the axioms. But to me - while often surprisingly hard to prove - these questions seem simple enough, in the sense that they only ask for the existence of some integers satisfying algebraic equations, meaning you could write a computer program checking through all tuples of integers $(a_1 , \ldots , a_n)$. Of course if there is no such tuple your program won't terminate, but in that case I would still find it ridiculous if you could not prove that there are none.
I don't see an argument for what I am suspecting which is why I am asking this question. I would appreciate any information on the matter because this has been confusing me for a while.

Answer 1

meaning you could write a computer program checking through all tuples of integers $(a_1 , \ldots , a_n)$.

You certainly could not! There are infinitely many of them! (That is to say, you could attempt to do this, and your program could run for hundreds of years without returning a result, and you still wouldn't know whether or not there were solutions.)

Anyway, the answer to your question is yes, suitably interpreted. This follows from Matiyasevich's theorem, which says that the sets of tuples of integers that can be represented as solutions to Diophantine equations (these equations include extra parameters and ask for the existence of those parameters, e.g. "there exists $n$ such that $x = n^2$" counts) are exactly the sets of tuples of integers which are recognizable by a Turing machine.

In particular, it follows that there is a specific (very large) Diophantine equation one can write down which has solutions if and only if a Turing machine searching for a proof of a contradiction in ZFC terminates; in other words, if and only if ZFC is inconsistent. So the existence of solutions to this Diophantine equation is independent of ZFC.