Minimum Steps to reach to a Smaller Number in 3n+1/Collatz Process
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866911399744110592 |
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| author | Sha, Daohang |
| author_facet | Sha, Daohang |
| contents | We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for the associated stopping and cycle numbers. These bounds imply that $F_{q}(m)/m \to 1$ as the sequence length increases, while equality is impossible for any finite sequence. Consequently, no finite nontrivial cycle is compatible with the iteration, and the trivial cycle at $1$ is the only admissible periodic orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_12962 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Minimum Steps to reach to a Smaller Number in 3n+1/Collatz Process Sha, Daohang General Mathematics We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for the associated stopping and cycle numbers. These bounds imply that $F_{q}(m)/m \to 1$ as the sequence length increases, while equality is impossible for any finite sequence. Consequently, no finite nontrivial cycle is compatible with the iteration, and the trivial cycle at $1$ is the only admissible periodic orbit. |
| title | Minimum Steps to reach to a Smaller Number in 3n+1/Collatz Process |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2112.12962 |