Resolution of the Collatz Conjecture via Structural and Quantitative Approaches

2025 · doi:10.22541/au.173660871.17716215/v1
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Abstract

We present a comprehensive resolution of the Collatz conjecture through two complementary approaches. The primary approach develops a novel framework based on inverse function properties and generative completeness, establishing that all trajectories in the Collatz dynamical system converge to the cycle {1 , 4 , 2}. A second, independent proof using measure theory provides explicit computational bounds for convergence times. Together, these approaches offer both structural understanding and quantitative characterization of the system’s behavior, providing a complete resolution of this long-standing problem.
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Abstract

We present a comprehensive resolution of the Collatz conjecture through two complementary approaches. The primary approach develops a novel framework based on inverse function properties and generative completeness, establishing that all trajectories in the Collatz dynamical system converge to the cycle {1 , 4 , 2}. A second, independent proof using measure theory provides explicit computational bounds for convergence times. Together, these approaches offer both structural understanding and quantitative characterization of the system’s behavior, providing a complete resolution of this long-standing problem. Supplementary Material File (collatz essential cinque.pdf) - Download - 332.48 KB Information & Authors Information Version history Copyright This work is licensed under a Non Exclusive No Reuse License.

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Authors Metrics & Citations Metrics Article Usage 441views 142downloads Citations Download citation Eduardo Diedrich. Resolution of the Collatz Conjecture via Structural and Quantitative Approaches. Authorea. 11 January 2025. DOI: https://doi.org/10.22541/au.173660871.17716215/v1 DOI: https://doi.org/10.22541/au.173660871.17716215/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu.

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