Theoretical analysis demonstrates finite orbit convergence to the 4-2-1 cycle in all positive integers, suggesting resolution of the Collatz conjecture.
The Collatz conjecture is a well‑known unsolved problem in number theory. In this paper, we perform an exhaustive analysis of the iteration orbits of all positive odd integers using a full‑classification method modulo 8. By a rigorous finite‑descent argument, we prove that the Collatz orbit of any positive integer must converge to the cycle (4,2,1) in finitely many steps. Consequently, the Collatz conjecture is completely proved.
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Fucheng Zhu (2026) studied this question.