Veryifying the Collatz Conjecture for one number without explicit computation

Question

Basically the title. Exhaustive computer searches are going on all the time, but if we seem to find a very large counterexample to the Collatz Conjecture, how will we ever know that it doesn't, in the end, hit $1$? Are there any results, theorems, or algorithms that can help us here?

Answer 1

"Are there any results, theorems, or algorithms that can help us here?" Perhaps the following is interesting if not helpful: if there really is a nontrivial cycle in the Collatz problem, then $\det M (d) = 0$ for all large $d$, but if there are none then $\det M (d) = (−1)^d$ for all $d$. Here $M(d)$ is Zeilberger's matrix.

References: Chapman

Zeilberger's determinant evaluation problem

Answer 2

You're touching upon the notion that the conjecture may be unprovable within certain mathematical systems. This does indeed follow from your observation that one counterexample could ascend indefinitely to infinity.

If a theory weren't strong enough, any proof that some infinitely long sequence exists might itself not be writable in a finite number of steps, making such a proof impossible to complete.