Iteration Steps of 3x+1 Problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908445627645952 |
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| author | Luo, Youchun |
| author_facet | Luo, Youchun |
| contents | On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $ be the total iteration steps, the odd iteration steps and the even iteration steps when $N$ iterates to 1(except 1) respectively. Trivially, we have $D\left( N \right) =O\left( N \right) +E\left( N \right) $. In this paper, we propose a so-called weak residue conjecture(i.e., $\frac{2^{E\left( N \right)}}{3^{O\left( N \right)}\cdot N}\le 2$). We prove that if 3x+1 conjecture is true and the weak residue conjecture is true, there exist non-trivial relationships among $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $, i.e., $O\left( N \right) =\lfloor \log_6 2\cdot D\left( N \right) -\log_6 N \rfloor $(it implies that we can calculate $O\left( N \right) $, $E\left( N \right) $ directly by $D\left( N \right) $ only, of course given $N$), and 5 more similar equations are derived simultaneously. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iteration Steps of 3x+1 Problem Luo, Youchun General Mathematics 11B83 On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $ be the total iteration steps, the odd iteration steps and the even iteration steps when $N$ iterates to 1(except 1) respectively. Trivially, we have $D\left( N \right) =O\left( N \right) +E\left( N \right) $. In this paper, we propose a so-called weak residue conjecture(i.e., $\frac{2^{E\left( N \right)}}{3^{O\left( N \right)}\cdot N}\le 2$). We prove that if 3x+1 conjecture is true and the weak residue conjecture is true, there exist non-trivial relationships among $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $, i.e., $O\left( N \right) =\lfloor \log_6 2\cdot D\left( N \right) -\log_6 N \rfloor $(it implies that we can calculate $O\left( N \right) $, $E\left( N \right) $ directly by $D\left( N \right) $ only, of course given $N$), and 5 more similar equations are derived simultaneously. |
| title | Iteration Steps of 3x+1 Problem |
| topic | General Mathematics 11B83 |
| url | https://arxiv.org/abs/2506.23070 |