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March 23, 20260 citationsOpen Access

Uniqueness of the Trivial Cycle in the Collatz Conjecture

Key Points

The research aims to prove that the only cycle in the Collatz sequence is the trivial cycle.
Algebraic reduction to a Diophantine equation
Numerical verification up to a computable boundary
Asymptotic analysis beyond the computable boundary
Confirmed that the only cycle is 1 → 4 → 2 → 1
Established that nontrivial cycles are impossible

Abstract

We prove that the only cycle in the Collatz sequence (the 3n+1 problem) is the trivial cycle 1 → 4 → 2 → 1. The proof combines algebraic reduction to a Diophantine equation, numerical verification up to a computable boundary, and asymptotic analysis beyond that boundary. This result establishes that nontrivial cycles in the Collatz conjecture are impossible.

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Vitaly Reznik (Sat,) studied this question.

synapsesocial.com/papers/69c0e051fddb9876e79c1cef https://doi.org/https://doi.org/10.5281/zenodo.19153136

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