A general question about the Collatz Conjecture and finding that integer that doesn't work

Solution 1:

Yes, you're right: If there are any counterexample, then the smallest counterexample must be odd. And all successors and predecessors of a counterexample are themselves counterexamples.

There are two conceivable kinds of counterexample.

The first is a finite cyclic sequence that differs from the trivial 1-4-2-1 cycle. Finding such a counterexample would immediately produce a disproof of the conjecture.

The second is a starting point from which the sequence continues indefinitely without ever hitting a cycle. Just coming across such a point would not directly yield a disproof of the conjecture, because one would need to prove that the sequence does indeed never join a cycle, and there's no known systematic way of findig such a proof.

Solution 2:

You are correct, in both your points. Suppose that the conjecture is false.

If $N$ is a counterexample and $M$ is an element of the sequence generated by $N$, then $M$ is a counterexample as well, since if the sequence generated by $M$ ended to $1$, then also the sequence generated by $N$ ends at $1$.

For the same reason, the minimal counterexample has to be odd, because if $N$ is an even counterexample, then $N/2$ is a counterexample as well, as it is the first term generated by the sequence starting with $N$.