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arXiv · 1908.00311

A Refinement of the $3x+1$ Conjecture

Roger Zarnowski
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Abstract

We reformulate the $3x+1$ conjecture by restricting attention to numbers congruent to $2$ (mod $3$). This leads to an equivalent conjecture for positive integers that reveals new aspects of the dynamics of the $3x+1$ problem. Advantages include a governing function with particularly simple mapping properties in terms of partitions of the set of integers. We use the refined conjecture to obtain a new characterization of $3x+1$ trajectories that shows a special role played by numbers congruent to $2$ or $8$ (mod $9$). We construct an accelerated iteration whose long-term behavior involves only those numbers.

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Roger Zarnowski. 2019-08-01. A Refinement of the $3x+1$ Conjecture. https://doi.org/10.7546/nntdm.2020.26.3.234-244

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