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Prove the Collatz Conjecture

Prove that for every positive integer n, the Collatz iteration defined by T(n) = n/2 when n is even and T(n) = 3n + 1 when n is odd eventually reaches 1, equivalently entering the cycle 4, 2, 1.

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Background

The paper studies computational aspects of the Collatz iteration and presents an algorithm that improves the efficiency of computing stopping times by leveraging structural patterns in the Collatz tree. Although the algorithm achieves about 28% computational improvement over existing methods and scales to very large inputs, the underlying mathematical question of universal convergence for the Collatz iteration remains unresolved.

Within the introduction, the authors explicitly characterize the Collatz conjecture as a long-standing open question. The remainder of the paper focuses on algorithmic computation of stopping times and empirical verification at large scales, but does not provide a proof of universal convergence, thereby underscoring the conjecture’s status as an open mathematical problem.

References

The Collatz conjecture, also known as the 3n+1 problem, is a long-standing open question in mathematics that continues to captivate researchers due to its simplicity and the profound computational challenges it poses. Proposed by Lothar Collatz in 1937, the conjecture posits that any positive integer will eventually reach the cycle 4, 2, 1 through a specific iterative process: dividing by 2 if the number is even, or applying the transformation 3n + 1 if odd.

— Efficient Computation of Collatz Sequence Stopping Times: A Novel Algorithmic Approach (2501.04032 - Getachew et al., 1 Jan 2025) in Section 1 (Introduction)