I need some clarification about what would prove that the conjecture holds for every positive integer. Some people claim that proving:
is not necessary and sufficient for the proof.
Are they right, or wrong?
If they are right, what are the necessary and sufficient conditions to accept an attempt as a proof?
If it can be rigorously proven that every natural number belongs to a rooted tree structure with 1 as its root, then this is sufficient to prove the Collatz conjecture.
Explanation:
Such a rooted tree structure implies that:
There are no cycles (loops),
Every vertex (natural number) has a unique path from the root (1), and
There are no isolated nodes disconnected from the root.
If these conditions hold, it guarantees that any natural number reaches 1 in a finite number of steps, and no nontrivial cycles exist.
Therefore:
Showing that all natural numbers are contained in a rooted tree with root 1 is a necessary and sufficient condition to prove the Collatz conjecture.
If my recent preview (Preprint Collatz Graph and Its Inverted Interpretation: A Tree-Based ...
) doesn't have any gaps or flows (I checked it several times before I published) then I have positively replied to ifs (see Section 3 of the paper).
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